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dynhpg.F90 in NEMO/trunk/src/OCE/DYN – NEMO

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source: NEMO/trunk/src/OCE/DYN/dynhpg.F90 @ 14433

Last change on this file since 14433 was 14433, checked in by smasson, 3 years ago
trunk: merge dev_r14312_MPI_Interface into the trunk, #2598
Property svn:keywords set to `Id`
File size: 75.4 KB

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1	MODULE dynhpg
2	!!======================================================================
3	!! * MODULE dynhpg *
4	!! Ocean dynamics: hydrostatic pressure gradient trend
5	!!======================================================================
6	!! History : OPA ! 1987-09 (P. Andrich, M.-A. Foujols) hpg_zco: Original code
7	!! 5.0 ! 1991-11 (G. Madec)
8	!! 7.0 ! 1996-01 (G. Madec) hpg_sco: Original code for s-coordinates
9	!! 8.0 ! 1997-05 (G. Madec) split dynber into dynkeg and dynhpg
10	!! 8.5 ! 2002-07 (G. Madec) F90: Free form and module
11	!! 8.5 ! 2002-08 (A. Bozec) hpg_zps: Original code
12	!! NEMO 1.0 ! 2005-10 (A. Beckmann, B.W. An) various s-coordinate options
13	!! ! Original code for hpg_ctl, hpg_hel hpg_wdj, hpg_djc, hpg_rot
14	!! - ! 2005-11 (G. Madec) style & small optimisation
15	!! 3.3 ! 2010-10 (C. Ethe, G. Madec) reorganisation of initialisation phase
16	!! 3.4 ! 2011-11 (H. Liu) hpg_prj: Original code for s-coordinates
17	!! ! (A. Coward) suppression of hel, wdj and rot options
18	!! 3.6 ! 2014-11 (P. Mathiot) hpg_isf: original code for ice shelf cavity
19	!! 4.2 ! 2020-12 (M. Bell, A. Young) hpg_djc: revised djc scheme
20	!!----------------------------------------------------------------------
21
22	!!----------------------------------------------------------------------
23	!! dyn_hpg : update the momentum trend with the now horizontal
24	!! gradient of the hydrostatic pressure
25	!! dyn_hpg_init : initialisation and control of options
26	!! hpg_zco : z-coordinate scheme
27	!! hpg_zps : z-coordinate plus partial steps (interpolation)
28	!! hpg_sco : s-coordinate (standard jacobian formulation)
29	!! hpg_isf : s-coordinate (sco formulation) adapted to ice shelf
30	!! hpg_djc : s-coordinate (Density Jacobian with Cubic polynomial)
31	!! hpg_prj : s-coordinate (Pressure Jacobian with Cubic polynomial)
32	!!----------------------------------------------------------------------
33	USE oce ! ocean dynamics and tracers
34	USE isf_oce , ONLY : risfload ! ice shelf (risfload variable)
35	USE isfload , ONLY : isf_load ! ice shelf (isf_load routine )
36	USE sbc_oce ! surface variable (only for the flag with ice shelf)
37	USE dom_oce ! ocean space and time domain
38	USE wet_dry ! wetting and drying
39	USE phycst ! physical constants
40	USE trd_oce ! trends: ocean variables
41	USE trddyn ! trend manager: dynamics
42	USE zpshde ! partial step: hor. derivative (zps_hde routine)
43	!
44	USE in_out_manager ! I/O manager
45	USE prtctl ! Print control
46	USE lbclnk ! lateral boundary condition
47	USE lib_mpp ! MPP library
48	USE eosbn2 ! compute density
49	USE timing ! Timing
50	USE iom
51
52	IMPLICIT NONE
53	PRIVATE
54
55	PUBLIC dyn_hpg ! routine called by step module
56	PUBLIC dyn_hpg_init ! routine called by opa module
57
58	! !!* Namelist namdyn_hpg : hydrostatic pressure gradient
59	LOGICAL, PUBLIC :: ln_hpg_zco !: z-coordinate - full steps
60	LOGICAL, PUBLIC :: ln_hpg_zps !: z-coordinate - partial steps (interpolation)
61	LOGICAL, PUBLIC :: ln_hpg_sco !: s-coordinate (standard jacobian formulation)
62	LOGICAL, PUBLIC :: ln_hpg_djc !: s-coordinate (Density Jacobian with Cubic polynomial)
63	LOGICAL, PUBLIC :: ln_hpg_prj !: s-coordinate (Pressure Jacobian scheme)
64	LOGICAL, PUBLIC :: ln_hpg_isf !: s-coordinate similar to sco modify for isf
65
66	! !! Flag to control the type of hydrostatic pressure gradient
67	INTEGER, PARAMETER :: np_ERROR =-10 ! error in specification of lateral diffusion
68	INTEGER, PARAMETER :: np_zco = 0 ! z-coordinate - full steps
69	INTEGER, PARAMETER :: np_zps = 1 ! z-coordinate - partial steps (interpolation)
70	INTEGER, PARAMETER :: np_sco = 2 ! s-coordinate (standard jacobian formulation)
71	INTEGER, PARAMETER :: np_djc = 3 ! s-coordinate (Density Jacobian with Cubic polynomial)
72	INTEGER, PARAMETER :: np_prj = 4 ! s-coordinate (Pressure Jacobian scheme)
73	INTEGER, PARAMETER :: np_isf = 5 ! s-coordinate similar to sco modify for isf
74	!
75	INTEGER, PUBLIC :: nhpg !: type of pressure gradient scheme used ! (deduced from ln_hpg_... flags) (PUBLIC for TAM)
76	!
77	LOGICAL :: ln_hpg_djc_vnh, ln_hpg_djc_vnv ! flag to specify hpg_djc boundary condition type
78	REAL(wp), PUBLIC :: aco_bc_hor, bco_bc_hor, aco_bc_vrt, bco_bc_vrt !: coefficients for hpg_djc hor and vert boundary conditions
79
80	!! * Substitutions
81	# include "do_loop_substitute.h90"
82	# include "domzgr_substitute.h90"
83
84	!!----------------------------------------------------------------------
85	!! NEMO/OCE 4.0 , NEMO Consortium (2018)
86	!! $Id$
87	!! Software governed by the CeCILL license (see ./LICENSE)
88	!!----------------------------------------------------------------------
89	CONTAINS
90
91	SUBROUTINE dyn_hpg( kt, Kmm, puu, pvv, Krhs )
92	!!---------------------------------------------------------------------
93	!! * ROUTINE dyn_hpg *
94	!!
95	!! ** Method : Call the hydrostatic pressure gradient routine
96	!! using the scheme defined in the namelist
97	!!
98	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
99	!! - send trends to trd_dyn for futher diagnostics (l_trddyn=T)
100	!!----------------------------------------------------------------------
101	INTEGER , INTENT( in ) :: kt ! ocean time-step index
102	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
103	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
104	!
105	REAL(wp), ALLOCATABLE, DIMENSION(:,:,:) :: ztrdu, ztrdv
106	!!----------------------------------------------------------------------
107	!
108	IF( ln_timing ) CALL timing_start('dyn_hpg')
109	!
110	IF( l_trddyn ) THEN ! Temporary saving of puu(:,:,:,Krhs) and pvv(:,:,:,Krhs) trends (l_trddyn)
111	ALLOCATE( ztrdu(jpi,jpj,jpk) , ztrdv(jpi,jpj,jpk) )
112	ztrdu(:,:,:) = puu(:,:,:,Krhs)
113	ztrdv(:,:,:) = pvv(:,:,:,Krhs)
114	ENDIF
115	!
116	SELECT CASE ( nhpg ) ! Hydrostatic pressure gradient computation
117	CASE ( np_zco ) ; CALL hpg_zco ( kt, Kmm, puu, pvv, Krhs ) ! z-coordinate
118	CASE ( np_zps ) ; CALL hpg_zps ( kt, Kmm, puu, pvv, Krhs ) ! z-coordinate plus partial steps (interpolation)
119	CASE ( np_sco ) ; CALL hpg_sco ( kt, Kmm, puu, pvv, Krhs ) ! s-coordinate (standard jacobian formulation)
120	CASE ( np_djc ) ; CALL hpg_djc ( kt, Kmm, puu, pvv, Krhs ) ! s-coordinate (Density Jacobian with Cubic polynomial)
121	CASE ( np_prj ) ; CALL hpg_prj ( kt, Kmm, puu, pvv, Krhs ) ! s-coordinate (Pressure Jacobian scheme)
122	CASE ( np_isf ) ; CALL hpg_isf ( kt, Kmm, puu, pvv, Krhs ) ! s-coordinate similar to sco modify for ice shelf
123	END SELECT
124	!
125	IF( l_trddyn ) THEN ! save the hydrostatic pressure gradient trends for momentum trend diagnostics
126	ztrdu(:,:,:) = puu(:,:,:,Krhs) - ztrdu(:,:,:)
127	ztrdv(:,:,:) = pvv(:,:,:,Krhs) - ztrdv(:,:,:)
128	CALL trd_dyn( ztrdu, ztrdv, jpdyn_hpg, kt, Kmm )
129	DEALLOCATE( ztrdu , ztrdv )
130	ENDIF
131	!
132	IF(sn_cfctl%l_prtctl) CALL prt_ctl( tab3d_1=puu(:,:,:,Krhs), clinfo1=' hpg - Ua: ', mask1=umask, &
133	& tab3d_2=pvv(:,:,:,Krhs), clinfo2= ' Va: ', mask2=vmask, clinfo3='dyn' )
134	!
135	IF( ln_timing ) CALL timing_stop('dyn_hpg')
136	!
137	END SUBROUTINE dyn_hpg
138
139
140	SUBROUTINE dyn_hpg_init( Kmm )
141	!!----------------------------------------------------------------------
142	!! * ROUTINE dyn_hpg_init *
143	!!
144	!! ** Purpose : initializations for the hydrostatic pressure gradient
145	!! computation and consistency control
146	!!
147	!! ** Action : Read the namelist namdyn_hpg and check the consistency
148	!! with the type of vertical coordinate used (zco, zps, sco)
149	!!----------------------------------------------------------------------
150	INTEGER, INTENT( in ) :: Kmm ! ocean time level index
151	!
152	INTEGER :: ioptio = 0 ! temporary integer
153	INTEGER :: ios ! Local integer output status for namelist read
154	!!
155	INTEGER :: ji, jj, jk, ikt ! dummy loop indices ISF
156	REAL(wp), ALLOCATABLE, DIMENSION(:,:,:) :: zts_top, zrhd ! hypothesys on isf density
157	REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zrhdtop_isf ! density at bottom of ISF
158	REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: ziceload ! density at bottom of ISF
159	!!
160	NAMELIST/namdyn_hpg/ ln_hpg_zco, ln_hpg_zps, ln_hpg_sco, &
161	& ln_hpg_djc, ln_hpg_prj, ln_hpg_isf, &
162	& ln_hpg_djc_vnh, ln_hpg_djc_vnv
163	!!----------------------------------------------------------------------
164	!
165	READ ( numnam_ref, namdyn_hpg, IOSTAT = ios, ERR = 901)
166	901 IF( ios /= 0 ) CALL ctl_nam ( ios , 'namdyn_hpg in reference namelist' )
167	!
168	READ ( numnam_cfg, namdyn_hpg, IOSTAT = ios, ERR = 902 )
169	902 IF( ios > 0 ) CALL ctl_nam ( ios , 'namdyn_hpg in configuration namelist' )
170	IF(lwm) WRITE ( numond, namdyn_hpg )
171	!
172	IF(lwp) THEN ! Control print
173	WRITE(numout,*)
174	WRITE(numout,*) 'dyn_hpg_init : hydrostatic pressure gradient initialisation'
175	WRITE(numout,*) '~~~~~~~~~~~~'
176	WRITE(numout,*) ' Namelist namdyn_hpg : choice of hpg scheme'
177	WRITE(numout,*) ' z-coord. - full steps ln_hpg_zco = ', ln_hpg_zco
178	WRITE(numout,*) ' z-coord. - partial steps (interpolation) ln_hpg_zps = ', ln_hpg_zps
179	WRITE(numout,*) ' s-coord. (standard jacobian formulation) ln_hpg_sco = ', ln_hpg_sco
180	WRITE(numout,*) ' s-coord. (standard jacobian formulation) for isf ln_hpg_isf = ', ln_hpg_isf
181	WRITE(numout,*) ' s-coord. (Density Jacobian: Cubic polynomial) ln_hpg_djc = ', ln_hpg_djc
182	WRITE(numout,*) ' s-coord. (Pressure Jacobian: Cubic polynomial) ln_hpg_prj = ', ln_hpg_prj
183	ENDIF
184	!
185	IF( .NOT.ln_linssh .AND. (ln_hpg_zco.OR.ln_hpg_zps) ) &
186	& CALL ctl_stop( 'dyn_hpg_init : non-linear free surface incompatible with hpg_zco or hpg_zps' )
187	!
188	IF( (.NOT.ln_hpg_isf .AND. ln_isfcav) .OR. (ln_hpg_isf .AND. .NOT.ln_isfcav) ) &
189	& CALL ctl_stop( 'dyn_hpg_init : ln_hpg_isf=T requires ln_isfcav=T and vice versa' )
190	!
191	#if defined key_qco
192	IF( ln_hpg_isf ) THEN
193	CALL ctl_stop( 'dyn_hpg_init : key_qco and ln_hpg_isf not yet compatible' )
194	ENDIF
195	#endif
196	!
197	! ! Set nhpg from ln_hpg_... flags & consistency check
198	nhpg = np_ERROR
199	ioptio = 0
200	IF( ln_hpg_zco ) THEN ; nhpg = np_zco ; ioptio = ioptio +1 ; ENDIF
201	IF( ln_hpg_zps ) THEN ; nhpg = np_zps ; ioptio = ioptio +1 ; ENDIF
202	IF( ln_hpg_sco ) THEN ; nhpg = np_sco ; ioptio = ioptio +1 ; ENDIF
203	IF( ln_hpg_djc ) THEN ; nhpg = np_djc ; ioptio = ioptio +1 ; ENDIF
204	IF( ln_hpg_prj ) THEN ; nhpg = np_prj ; ioptio = ioptio +1 ; ENDIF
205	IF( ln_hpg_isf ) THEN ; nhpg = np_isf ; ioptio = ioptio +1 ; ENDIF
206	!
207	IF( ioptio /= 1 ) CALL ctl_stop( 'NO or several hydrostatic pressure gradient options used' )
208	!
209	IF(lwp) THEN
210	WRITE(numout,*)
211	SELECT CASE( nhpg )
212	CASE( np_zco ) ; WRITE(numout,*) ' ==>>> z-coord. - full steps '
213	CASE( np_zps ) ; WRITE(numout,*) ' ==>>> z-coord. - partial steps (interpolation)'
214	CASE( np_sco ) ; WRITE(numout,*) ' ==>>> s-coord. (standard jacobian formulation)'
215	CASE( np_djc ) ; WRITE(numout,*) ' ==>>> s-coord. (Density Jacobian: Cubic polynomial)'
216	CASE( np_prj ) ; WRITE(numout,*) ' ==>>> s-coord. (Pressure Jacobian: Cubic polynomial)'
217	CASE( np_isf ) ; WRITE(numout,*) ' ==>>> s-coord. (standard jacobian formulation) for isf'
218	END SELECT
219	WRITE(numout,*)
220	ENDIF
221	!
222	IF ( ln_hpg_djc ) THEN
223	IF (ln_hpg_djc_vnh) THEN ! Von Neumann boundary condition
224	IF(lwp) WRITE(numout,*) ' horizontal bc: von Neumann '
225	aco_bc_hor = 6.0_wp/5.0_wp
226	bco_bc_hor = 7.0_wp/15.0_wp
227	ELSE ! Linear extrapolation
228	IF(lwp) WRITE(numout,*) ' horizontal bc: linear extrapolation'
229	aco_bc_hor = 3.0_wp/2.0_wp
230	bco_bc_hor = 1.0_wp/2.0_wp
231	END IF
232	IF (ln_hpg_djc_vnv) THEN ! Von Neumann boundary condition
233	IF(lwp) WRITE(numout,*) ' vertical bc: von Neumann '
234	aco_bc_vrt = 6.0_wp/5.0_wp
235	bco_bc_vrt = 7.0_wp/15.0_wp
236	ELSE ! Linear extrapolation
237	IF(lwp) WRITE(numout,*) ' vertical bc: linear extrapolation'
238	aco_bc_vrt = 3.0_wp/2.0_wp
239	bco_bc_vrt = 1.0_wp/2.0_wp
240	END IF
241	END IF
242	!
243	END SUBROUTINE dyn_hpg_init
244
245
246	SUBROUTINE hpg_zco( kt, Kmm, puu, pvv, Krhs )
247	!!---------------------------------------------------------------------
248	!! * ROUTINE hpg_zco *
249	!!
250	!! ** Method : z-coordinate case, levels are horizontal surfaces.
251	!! The now hydrostatic pressure gradient at a given level, jk,
252	!! is computed by taking the vertical integral of the in-situ
253	!! density gradient along the model level from the suface to that
254	!! level: zhpi = grav .....
255	!! zhpj = grav .....
256	!! add it to the general momentum trend (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)).
257	!! puu(:,:,:,Krhs) = puu(:,:,:,Krhs) - 1/e1u * zhpi
258	!! pvv(:,:,:,Krhs) = pvv(:,:,:,Krhs) - 1/e2v * zhpj
259	!!
260	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
261	!!----------------------------------------------------------------------
262	INTEGER , INTENT( in ) :: kt ! ocean time-step index
263	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
264	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
265	!
266	INTEGER :: ji, jj, jk ! dummy loop indices
267	REAL(wp) :: zcoef0, zcoef1 ! temporary scalars
268	REAL(wp), DIMENSION(jpi,jpj) :: zhpi, zhpj
269	!!----------------------------------------------------------------------
270	!
271	IF( kt == nit000 ) THEN
272	IF(lwp) WRITE(numout,*)
273	IF(lwp) WRITE(numout,*) 'dyn:hpg_zco : hydrostatic pressure gradient trend'
274	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ z-coordinate case '
275	ENDIF
276	!
277	zcoef0 = - grav * 0.5_wp ! Local constant initialization
278	!
279	DO_2D( 0, 0, 0, 0 ) ! Surface value
280	zcoef1 = zcoef0 * e3w(ji,jj,1,Kmm)
281	! ! hydrostatic pressure gradient
282	zhpi(ji,jj) = zcoef1 * ( rhd(ji+1,jj,1) - rhd(ji,jj,1) ) * r1_e1u(ji,jj)
283	zhpj(ji,jj) = zcoef1 * ( rhd(ji,jj+1,1) - rhd(ji,jj,1) ) * r1_e2v(ji,jj)
284	! ! add to the general momentum trend
285	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + zhpi(ji,jj)
286	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + zhpj(ji,jj)
287	END_2D
288	!
289	DO_3D( 0, 0, 0, 0, 2, jpkm1 ) ! interior value (2=<jk=<jpkm1)
290	zcoef1 = zcoef0 * e3w(ji,jj,jk,Kmm)
291	! ! hydrostatic pressure gradient
292	zhpi(ji,jj) = zhpi(ji,jj) + zcoef1 * ( ( rhd(ji+1,jj,jk)+rhd(ji+1,jj,jk-1) ) &
293	& - ( rhd(ji ,jj,jk)+rhd(ji ,jj,jk-1) ) ) * r1_e1u(ji,jj)
294
295	zhpj(ji,jj) = zhpj(ji,jj) + zcoef1 * ( ( rhd(ji,jj+1,jk)+rhd(ji,jj+1,jk-1) ) &
296	& - ( rhd(ji,jj, jk)+rhd(ji,jj ,jk-1) ) ) * r1_e2v(ji,jj)
297	! ! add to the general momentum trend
298	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + zhpi(ji,jj)
299	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + zhpj(ji,jj)
300	END_3D
301	!
302	END SUBROUTINE hpg_zco
303
304
305	SUBROUTINE hpg_zps( kt, Kmm, puu, pvv, Krhs )
306	!!---------------------------------------------------------------------
307	!! * ROUTINE hpg_zps *
308	!!
309	!! ** Method : z-coordinate plus partial steps case. blahblah...
310	!!
311	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
312	!!----------------------------------------------------------------------
313	INTEGER , INTENT( in ) :: kt ! ocean time-step index
314	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
315	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
316	!!
317	INTEGER :: ji, jj, jk ! dummy loop indices
318	INTEGER :: iku, ikv ! temporary integers
319	REAL(wp) :: zcoef0, zcoef1, zcoef2, zcoef3 ! temporary scalars
320	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zhpj
321	REAL(wp), DIMENSION(jpi,jpj,jpts) :: zgtsu, zgtsv
322	REAL(wp), DIMENSION(jpi,jpj) :: zgru, zgrv
323	!!----------------------------------------------------------------------
324	!
325	IF( kt == nit000 ) THEN
326	IF(lwp) WRITE(numout,*)
327	IF(lwp) WRITE(numout,*) 'dyn:hpg_zps : hydrostatic pressure gradient trend'
328	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ z-coordinate with partial steps - vector optimization'
329	ENDIF
330
331	! Partial steps: Compute NOW horizontal gradient of t, s, rd at the last ocean level
332	CALL zps_hde( kt, Kmm, jpts, ts(:,:,:,:,Kmm), zgtsu, zgtsv, rhd, zgru , zgrv )
333
334	! Local constant initialization
335	zcoef0 = - grav * 0.5_wp
336
337	! Surface value (also valid in partial step case)
338	DO_2D( 0, 0, 0, 0 )
339	zcoef1 = zcoef0 * e3w(ji,jj,1,Kmm)
340	! hydrostatic pressure gradient
341	zhpi(ji,jj,1) = zcoef1 * ( rhd(ji+1,jj ,1) - rhd(ji,jj,1) ) * r1_e1u(ji,jj)
342	zhpj(ji,jj,1) = zcoef1 * ( rhd(ji ,jj+1,1) - rhd(ji,jj,1) ) * r1_e2v(ji,jj)
343	! add to the general momentum trend
344	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + zhpi(ji,jj,1)
345	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + zhpj(ji,jj,1)
346	END_2D
347
348	! interior value (2=<jk=<jpkm1)
349	DO_3D( 0, 0, 0, 0, 2, jpkm1 )
350	zcoef1 = zcoef0 * e3w(ji,jj,jk,Kmm)
351	! hydrostatic pressure gradient
352	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
353	& + zcoef1 * ( ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) ) &
354	& - ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) ) ) * r1_e1u(ji,jj)
355
356	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
357	& + zcoef1 * ( ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) ) &
358	& - ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) ) ) * r1_e2v(ji,jj)
359	! add to the general momentum trend
360	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + zhpi(ji,jj,jk)
361	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + zhpj(ji,jj,jk)
362	END_3D
363
364	! partial steps correction at the last level (use zgru & zgrv computed in zpshde.F90)
365	DO_2D( 0, 0, 0, 0 )
366	iku = mbku(ji,jj)
367	ikv = mbkv(ji,jj)
368	zcoef2 = zcoef0 * MIN( e3w(ji,jj,iku,Kmm), e3w(ji+1,jj ,iku,Kmm) )
369	zcoef3 = zcoef0 * MIN( e3w(ji,jj,ikv,Kmm), e3w(ji ,jj+1,ikv,Kmm) )
370	IF( iku > 1 ) THEN ! on i-direction (level 2 or more)
371	puu (ji,jj,iku,Krhs) = puu(ji,jj,iku,Krhs) - zhpi(ji,jj,iku) ! subtract old value
372	zhpi(ji,jj,iku) = zhpi(ji,jj,iku-1) & ! compute the new one
373	& + zcoef2 * ( rhd(ji+1,jj,iku-1) - rhd(ji,jj,iku-1) + zgru(ji,jj) ) * r1_e1u(ji,jj)
374	puu (ji,jj,iku,Krhs) = puu(ji,jj,iku,Krhs) + zhpi(ji,jj,iku) ! add the new one to the general momentum trend
375	ENDIF
376	IF( ikv > 1 ) THEN ! on j-direction (level 2 or more)
377	pvv (ji,jj,ikv,Krhs) = pvv(ji,jj,ikv,Krhs) - zhpj(ji,jj,ikv) ! subtract old value
378	zhpj(ji,jj,ikv) = zhpj(ji,jj,ikv-1) & ! compute the new one
379	& + zcoef3 * ( rhd(ji,jj+1,ikv-1) - rhd(ji,jj,ikv-1) + zgrv(ji,jj) ) * r1_e2v(ji,jj)
380	pvv (ji,jj,ikv,Krhs) = pvv(ji,jj,ikv,Krhs) + zhpj(ji,jj,ikv) ! add the new one to the general momentum trend
381	ENDIF
382	END_2D
383	!
384	END SUBROUTINE hpg_zps
385
386
387	SUBROUTINE hpg_sco( kt, Kmm, puu, pvv, Krhs )
388	!!---------------------------------------------------------------------
389	!! * ROUTINE hpg_sco *
390	!!
391	!! ** Method : s-coordinate case. Jacobian scheme.
392	!! The now hydrostatic pressure gradient at a given level, jk,
393	!! is computed by taking the vertical integral of the in-situ
394	!! density gradient along the model level from the suface to that
395	!! level. s-coordinates (ln_sco): a corrective term is added
396	!! to the horizontal pressure gradient :
397	!! zhpi = grav ..... + 1/e1u mi(rhd) di[ grav dep3w ]
398	!! zhpj = grav ..... + 1/e2v mj(rhd) dj[ grav dep3w ]
399	!! add it to the general momentum trend (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)).
400	!! puu(:,:,:,Krhs) = puu(:,:,:,Krhs) - 1/e1u * zhpi
401	!! pvv(:,:,:,Krhs) = pvv(:,:,:,Krhs) - 1/e2v * zhpj
402	!!
403	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
404	!!----------------------------------------------------------------------
405	INTEGER , INTENT( in ) :: kt ! ocean time-step index
406	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
407	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
408	!!
409	INTEGER :: ji, jj, jk, jii, jjj ! dummy loop indices
410	REAL(wp) :: zcoef0, zuap, zvap, ztmp ! local scalars
411	LOGICAL :: ll_tmp1, ll_tmp2 ! local logical variables
412	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zhpj
413	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: zcpx, zcpy !W/D pressure filter
414	!!----------------------------------------------------------------------
415	!
416	IF( ln_wd_il ) ALLOCATE(zcpx(jpi,jpj), zcpy(jpi,jpj))
417	!
418	IF( kt == nit000 ) THEN
419	IF(lwp) WRITE(numout,*)
420	IF(lwp) WRITE(numout,*) 'dyn:hpg_sco : hydrostatic pressure gradient trend'
421	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, OCE original scheme used'
422	ENDIF
423	!
424	zcoef0 = - grav * 0.5_wp
425	!
426	IF( ln_wd_il ) THEN
427	DO_2D( 0, 0, 0, 0 )
428	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
429	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) .AND. &
430	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) ) &
431	& > rn_wdmin1 + rn_wdmin2
432	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji+1,jj,Kmm) ) > 1.E-12 ) .AND. ( &
433	& MAX( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
434	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) + rn_wdmin1 + rn_wdmin2 )
435
436	IF(ll_tmp1) THEN
437	zcpx(ji,jj) = 1.0_wp
438	ELSE IF(ll_tmp2) THEN
439	! no worries about ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm) = 0, it won't happen ! here
440	zcpx(ji,jj) = ABS( (ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
441	& / (ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm)) )
442	ELSE
443	zcpx(ji,jj) = 0._wp
444	END IF
445
446	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
447	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) .AND. &
448	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) ) &
449	& > rn_wdmin1 + rn_wdmin2
450	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji,jj+1,Kmm) ) > 1.E-12 ) .AND. ( &
451	& MAX( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
452	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) + rn_wdmin1 + rn_wdmin2 )
453
454	IF(ll_tmp1) THEN
455	zcpy(ji,jj) = 1.0_wp
456	ELSE IF(ll_tmp2) THEN
457	! no worries about ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm) = 0, it won't happen ! here
458	zcpy(ji,jj) = ABS( (ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
459	& / (ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm)) )
460	ELSE
461	zcpy(ji,jj) = 0._wp
462	END IF
463	END_2D
464	CALL lbc_lnk( 'dynhpg', zcpx, 'U', 1.0_wp, zcpy, 'V', 1.0_wp )
465	END IF
466	!
467	DO_2D( 0, 0, 0, 0 ) ! Surface value
468	! ! hydrostatic pressure gradient along s-surfaces
469	zhpi(ji,jj,1) = zcoef0 * r1_e1u(ji,jj) &
470	& * ( e3w(ji+1,jj ,1,Kmm) * rhd(ji+1,jj ,1) &
471	& - e3w(ji ,jj ,1,Kmm) * rhd(ji ,jj ,1) )
472	zhpj(ji,jj,1) = zcoef0 * r1_e2v(ji,jj) &
473	& * ( e3w(ji ,jj+1,1,Kmm) * rhd(ji ,jj+1,1) &
474	& - e3w(ji ,jj ,1,Kmm) * rhd(ji ,jj ,1) )
475	! ! s-coordinate pressure gradient correction
476	zuap = -zcoef0 * ( rhd (ji+1,jj,1) + rhd (ji,jj,1) ) &
477	& * ( gde3w(ji+1,jj,1) - gde3w(ji,jj,1) ) * r1_e1u(ji,jj)
478	zvap = -zcoef0 * ( rhd (ji,jj+1,1) + rhd (ji,jj,1) ) &
479	& * ( gde3w(ji,jj+1,1) - gde3w(ji,jj,1) ) * r1_e2v(ji,jj)
480	!
481	IF( ln_wd_il ) THEN
482	zhpi(ji,jj,1) = zhpi(ji,jj,1) * zcpx(ji,jj)
483	zhpj(ji,jj,1) = zhpj(ji,jj,1) * zcpy(ji,jj)
484	zuap = zuap * zcpx(ji,jj)
485	zvap = zvap * zcpy(ji,jj)
486	ENDIF
487	! ! add to the general momentum trend
488	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + zhpi(ji,jj,1) + zuap
489	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + zhpj(ji,jj,1) + zvap
490	END_2D
491	!
492	DO_3D( 0, 0, 0, 0, 2, jpkm1 ) ! interior value (2=<jk=<jpkm1)
493	! ! hydrostatic pressure gradient along s-surfaces
494	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + zcoef0 * r1_e1u(ji,jj) &
495	& * ( e3w(ji+1,jj,jk,Kmm) * ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) ) &
496	& - e3w(ji ,jj,jk,Kmm) * ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) ) )
497	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) + zcoef0 * r1_e2v(ji,jj) &
498	& * ( e3w(ji,jj+1,jk,Kmm) * ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) ) &
499	& - e3w(ji,jj ,jk,Kmm) * ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) ) )
500	! ! s-coordinate pressure gradient correction
501	zuap = -zcoef0 * ( rhd (ji+1,jj ,jk) + rhd (ji,jj,jk) ) &
502	& * ( gde3w(ji+1,jj ,jk) - gde3w(ji,jj,jk) ) * r1_e1u(ji,jj)
503	zvap = -zcoef0 * ( rhd (ji ,jj+1,jk) + rhd (ji,jj,jk) ) &
504	& * ( gde3w(ji ,jj+1,jk) - gde3w(ji,jj,jk) ) * r1_e2v(ji,jj)
505	!
506	IF( ln_wd_il ) THEN
507	zhpi(ji,jj,jk) = zhpi(ji,jj,jk) * zcpx(ji,jj)
508	zhpj(ji,jj,jk) = zhpj(ji,jj,jk) * zcpy(ji,jj)
509	zuap = zuap * zcpx(ji,jj)
510	zvap = zvap * zcpy(ji,jj)
511	ENDIF
512	!
513	! add to the general momentum trend
514	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + zhpi(ji,jj,jk) + zuap
515	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + zhpj(ji,jj,jk) + zvap
516	END_3D
517	!
518	IF( ln_wd_il ) DEALLOCATE( zcpx , zcpy )
519	!
520	END SUBROUTINE hpg_sco
521
522
523	SUBROUTINE hpg_isf( kt, Kmm, puu, pvv, Krhs )
524	!!---------------------------------------------------------------------
525	!! * ROUTINE hpg_isf *
526	!!
527	!! ** Method : s-coordinate case. Jacobian scheme.
528	!! The now hydrostatic pressure gradient at a given level, jk,
529	!! is computed by taking the vertical integral of the in-situ
530	!! density gradient along the model level from the suface to that
531	!! level. s-coordinates (ln_sco): a corrective term is added
532	!! to the horizontal pressure gradient :
533	!! zhpi = grav ..... + 1/e1u mi(rhd) di[ grav dep3w ]
534	!! zhpj = grav ..... + 1/e2v mj(rhd) dj[ grav dep3w ]
535	!! add it to the general momentum trend (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)).
536	!! puu(:,:,:,Krhs) = puu(:,:,:,Krhs) - 1/e1u * zhpi
537	!! pvv(:,:,:,Krhs) = pvv(:,:,:,Krhs) - 1/e2v * zhpj
538	!! iceload is added
539	!!
540	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
541	!!----------------------------------------------------------------------
542	INTEGER , INTENT( in ) :: kt ! ocean time-step index
543	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
544	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
545	!!
546	INTEGER :: ji, jj, jk ! dummy loop indices
547	INTEGER :: ikt , ikti1, iktj1 ! local integer
548	REAL(wp) :: ze3w, ze3wi1, ze3wj1 ! local scalars
549	REAL(wp) :: zcoef0, zuap, zvap ! - -
550	REAL(wp), DIMENSION(jpi,jpj,jpk ) :: zhpi, zhpj
551	REAL(wp), DIMENSION(jpi,jpj,jpts) :: zts_top
552	REAL(wp), DIMENSION(jpi,jpj) :: zrhdtop_oce
553	!!----------------------------------------------------------------------
554	!
555	zcoef0 = - grav * 0.5_wp ! Local constant initialization
556	!
557	! ! iniitialised to 0. zhpi zhpi
558	zhpi(:,:,:) = 0._wp ; zhpj(:,:,:) = 0._wp
559
560	! compute rhd at the ice/oce interface (ocean side)
561	! usefull to reduce residual current in the test case ISOMIP with no melting
562	DO ji = 1, jpi
563	DO jj = 1, jpj
564	ikt = mikt(ji,jj)
565	zts_top(ji,jj,1) = ts(ji,jj,ikt,1,Kmm)
566	zts_top(ji,jj,2) = ts(ji,jj,ikt,2,Kmm)
567	END DO
568	END DO
569	CALL eos( zts_top, risfdep, zrhdtop_oce )
570
571	! !===========================!
572	! !===== surface value =====!
573	! !===========================!
574	DO_2D( 0, 0, 0, 0 )
575	ikt = mikt(ji ,jj ) ; ze3w = e3w(ji ,jj ,ikt ,Kmm)
576	ikti1 = mikt(ji+1,jj ) ; ze3wi1 = e3w(ji+1,jj ,ikti1,Kmm)
577	iktj1 = mikt(ji ,jj+1) ; ze3wj1 = e3w(ji ,jj+1,iktj1,Kmm)
578	! ! hydrostatic pressure gradient along s-surfaces and ice shelf pressure
579	! ! we assume ISF is in isostatic equilibrium
580	zhpi(ji,jj,1) = zcoef0 * r1_e1u(ji,jj) * ( risfload(ji+1,jj) - risfload(ji,jj) &
581	& + 0.5_wp * ( ze3wi1 * ( rhd(ji+1,jj,ikti1) + zrhdtop_oce(ji+1,jj) ) &
582	& - ze3w * ( rhd(ji ,jj,ikt ) + zrhdtop_oce(ji ,jj) ) ) )
583	zhpj(ji,jj,1) = zcoef0 * r1_e2v(ji,jj) * ( risfload(ji,jj+1) - risfload(ji,jj) &
584	& + 0.5_wp * ( ze3wj1 * ( rhd(ji,jj+1,iktj1) + zrhdtop_oce(ji,jj+1) ) &
585	& - ze3w * ( rhd(ji,jj ,ikt ) + zrhdtop_oce(ji,jj ) ) ) )
586	! ! s-coordinate pressure gradient correction (=0 if z coordinate)
587	zuap = -zcoef0 * ( rhd (ji+1,jj,1) + rhd (ji,jj,1) ) &
588	& * ( gde3w(ji+1,jj,1) - gde3w(ji,jj,1) ) * r1_e1u(ji,jj)
589	zvap = -zcoef0 * ( rhd (ji,jj+1,1) + rhd (ji,jj,1) ) &
590	& * ( gde3w(ji,jj+1,1) - gde3w(ji,jj,1) ) * r1_e2v(ji,jj)
591	! ! add to the general momentum trend
592	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + (zhpi(ji,jj,1) + zuap) * umask(ji,jj,1)
593	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + (zhpj(ji,jj,1) + zvap) * vmask(ji,jj,1)
594	END_2D
595	!
596	! !=============================!
597	! !===== interior values =====!
598	! !=============================!
599	DO_3D( 0, 0, 0, 0, 2, jpkm1 )
600	ze3w = e3w(ji ,jj ,jk,Kmm)
601	ze3wi1 = e3w(ji+1,jj ,jk,Kmm)
602	ze3wj1 = e3w(ji ,jj+1,jk,Kmm)
603	! ! hydrostatic pressure gradient along s-surfaces
604	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + zcoef0 / e1u(ji,jj) &
605	& * ( ze3wi1 * ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) ) * wmask(ji+1,jj,jk) &
606	& - ze3w * ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) ) * wmask(ji ,jj,jk) )
607	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) + zcoef0 / e2v(ji,jj) &
608	& * ( ze3wj1 * ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) ) * wmask(ji,jj+1,jk) &
609	& - ze3w * ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) ) * wmask(ji,jj ,jk) )
610	! ! s-coordinate pressure gradient correction
611	zuap = -zcoef0 * ( rhd (ji+1,jj ,jk) + rhd (ji,jj,jk) ) &
612	& * ( gde3w(ji+1,jj ,jk) - gde3w(ji,jj,jk) ) / e1u(ji,jj)
613	zvap = -zcoef0 * ( rhd (ji ,jj+1,jk) + rhd (ji,jj,jk) ) &
614	& * ( gde3w(ji ,jj+1,jk) - gde3w(ji,jj,jk) ) / e2v(ji,jj)
615	! ! add to the general momentum trend
616	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + (zhpi(ji,jj,jk) + zuap) * umask(ji,jj,jk)
617	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + (zhpj(ji,jj,jk) + zvap) * vmask(ji,jj,jk)
618	END_3D
619	!
620	END SUBROUTINE hpg_isf
621
622
623	SUBROUTINE hpg_djc( kt, Kmm, puu, pvv, Krhs )
624	!!---------------------------------------------------------------------
625	!! * ROUTINE hpg_djc *
626	!!
627	!! ** Method : Density Jacobian with Cubic polynomial scheme
628	!!
629	!! Reference: Shchepetkin and McWilliams, J. Geophys. Res., 108(C3), 3090, 2003
630	!!----------------------------------------------------------------------
631	INTEGER , INTENT( in ) :: kt ! ocean time-step index
632	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
633	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
634	!!
635	INTEGER :: ji, jj, jk ! dummy loop indices
636	INTEGER :: iktb, iktt ! jk indices at tracer points for top and bottom points
637	REAL(wp) :: zcoef0, zep, cffw ! temporary scalars
638	REAL(wp) :: z_grav_10, z1_12
639	REAL(wp) :: cffu, cffx ! " "
640	REAL(wp) :: cffv, cffy ! " "
641	LOGICAL :: ll_tmp1, ll_tmp2 ! local logical variables
642	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zhpj
643
644	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zdzx, zdzy, zdzz ! Primitive grid differences ('delta_xyz')
645	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zdz_i, zdz_j, zdz_k ! Harmonic average of primitive grid differences ('d_xyz')
646	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zdrhox, zdrhoy, zdrhoz ! Primitive rho differences ('delta_rho')
647	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zdrho_i, zdrho_j, zdrho_k ! Harmonic average of primitive rho differences ('d_rho')
648	REAL(wp), DIMENSION(jpi,jpj,jpk) :: z_rho_i, z_rho_j, z_rho_k ! Face intergrals
649	REAL(wp), DIMENSION(jpi,jpj) :: zz_dz_i, zz_dz_j, zz_drho_i, zz_drho_j ! temporary arrays
650	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: zcpx, zcpy !W/D pressure filter
651	!!----------------------------------------------------------------------
652	!
653	IF( ln_wd_il ) THEN
654	ALLOCATE( zcpx(jpi,jpj) , zcpy(jpi,jpj) )
655	DO_2D( 0, 0, 0, 0 )
656	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
657	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) .AND. &
658	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) ) &
659	& > rn_wdmin1 + rn_wdmin2
660	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji+1,jj,Kmm) ) > 1.E-12 ) .AND. ( &
661	& MAX( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
662	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) + rn_wdmin1 + rn_wdmin2 )
663	IF(ll_tmp1) THEN
664	zcpx(ji,jj) = 1.0_wp
665	ELSE IF(ll_tmp2) THEN
666	! no worries about ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm) = 0, it won't happen ! here
667	zcpx(ji,jj) = ABS( (ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
668	& / (ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm)) )
669	ELSE
670	zcpx(ji,jj) = 0._wp
671	END IF
672
673	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
674	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) .AND. &
675	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) ) &
676	& > rn_wdmin1 + rn_wdmin2
677	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji,jj+1,Kmm) ) > 1.E-12 ) .AND. ( &
678	& MAX( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
679	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) + rn_wdmin1 + rn_wdmin2 )
680
681	IF(ll_tmp1) THEN
682	zcpy(ji,jj) = 1.0_wp
683	ELSE IF(ll_tmp2) THEN
684	! no worries about ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm) = 0, it won't happen ! here
685	zcpy(ji,jj) = ABS( (ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
686	& / (ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm)) )
687	ELSE
688	zcpy(ji,jj) = 0._wp
689	END IF
690	END_2D
691	CALL lbc_lnk( 'dynhpg', zcpx, 'U', 1.0_wp, zcpy, 'V', 1.0_wp )
692	END IF
693
694	IF( kt == nit000 ) THEN
695	IF(lwp) WRITE(numout,*)
696	IF(lwp) WRITE(numout,*) 'dyn:hpg_djc : hydrostatic pressure gradient trend'
697	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, density Jacobian with cubic polynomial scheme'
698	ENDIF
699
700	! Local constant initialization
701	zcoef0 = - grav * 0.5_wp
702	z_grav_10 = grav / 10._wp
703	z1_12 = 1.0_wp / 12._wp
704
705	!----------------------------------------------------------------------------------------
706	! 1. compute and store elementary vertical differences in provisional arrays
707	!----------------------------------------------------------------------------------------
708
709	!!bug gm Not a true bug, but... zdzz=e3w for zdzx, zdzy verify what it is really
710
711	DO_3D( 1, 1, 1, 1, 2, jpkm1 )
712	zdrhoz(ji,jj,jk) = rhd (ji ,jj ,jk) - rhd (ji,jj,jk-1)
713	zdzz (ji,jj,jk) = - gde3w(ji ,jj ,jk) + gde3w(ji,jj,jk-1)
714	END_3D
715
716	!-------------------------------------------------------------------------
717	! 2. compute harmonic averages for vertical differences using eq. 5.18
718	!-------------------------------------------------------------------------
719	zep = 1.e-15
720
721	!! mb zdrho_k, zdz_k, zdrho_i, zdz_i, zdrho_j, zdz_j re-centred about the point (ji,jj,jk)
722	zdrho_k(:,:,:) = 0._wp
723	zdz_k (:,:,:) = 0._wp
724
725	DO_3D( 1, 1, 1, 1, 2, jpk-2 )
726	cffw = 2._wp * zdrhoz(ji ,jj ,jk) * zdrhoz(ji,jj,jk+1)
727	IF( cffw > zep) THEN
728	zdrho_k(ji,jj,jk) = cffw / ( zdrhoz(ji,jj,jk) + zdrhoz(ji,jj,jk+1) )
729	ENDIF
730	zdz_k(ji,jj,jk) = 2._wp * zdzz(ji,jj,jk) * zdzz(ji,jj,jk+1) &
731	& / ( zdzz(ji,jj,jk) + zdzz(ji,jj,jk+1) )
732	END_3D
733
734	!----------------------------------------------------------------------------------
735	! 3. apply boundary conditions at top and bottom using 5.36-5.37
736	!----------------------------------------------------------------------------------
737
738	! mb for sea-ice shelves we will need to re-write this upper boundary condition in the same form as the lower boundary condition
739	zdrho_k(:,:,1) = aco_bc_vrt * ( rhd (:,:,2) - rhd (:,:,1) ) - bco_bc_vrt * zdrho_k(:,:,2)
740	zdz_k (:,:,1) = aco_bc_vrt * (-gde3w(:,:,2) + gde3w(:,:,1) ) - bco_bc_vrt * zdz_k (:,:,2)
741
742	DO_2D( 1, 1, 1, 1 )
743	IF ( mbkt(ji,jj)>1 ) THEN
744	iktb = mbkt(ji,jj)
745	zdrho_k(ji,jj,iktb) = aco_bc_vrt * ( rhd(ji,jj,iktb) - rhd(ji,jj,iktb-1) ) - bco_bc_vrt * zdrho_k(ji,jj,iktb-1)
746	zdz_k (ji,jj,iktb) = aco_bc_vrt * (-gde3w(ji,jj,iktb) + gde3w(ji,jj,iktb-1) ) - bco_bc_vrt * zdz_k (ji,jj,iktb-1)
747	END IF
748	END_2D
749
750	!--------------------------------------------------------------
751	! 4. Compute side face integrals
752	!-------------------------------------------------------------
753
754	!! ssh replaces e3w_n ; gde3w is a depth; the formulae involve heights
755	!! rho_k stores grav * FX / rho_0
756
757	!--------------------------------------------------------------
758	! 4. a) Upper half of top-most grid box, compute and store
759	!-------------------------------------------------------------
760	! *** AY note: ssh(ji,jj,Kmm) + gde3w(ji,jj,1) = e3w(ji,jj,1,Kmm)
761	DO_2D( 0, 1, 0, 1)
762	z_rho_k(ji,jj,1) = grav * ( ssh(ji,jj,Kmm) + gde3w(ji,jj,1) ) &
763	& * ( rhd(ji,jj,1) &
764	& + 0.5_wp * ( rhd (ji,jj,2) - rhd (ji,jj,1) ) &
765	& * ( ssh (ji,jj,Kmm) + gde3w(ji,jj,1) ) &
766	& / ( - gde3w(ji,jj,2) + gde3w(ji,jj,1) ) )
767	END_2D
768
769	!--------------------------------------------------------------
770	! 4. b) Interior faces, compute and store
771	!-------------------------------------------------------------
772
773	DO_3D( 0, 1, 0, 1, 2, jpkm1 )
774	z_rho_k(ji,jj,jk) = zcoef0 * ( rhd (ji,jj,jk) + rhd (ji,jj,jk-1) ) &
775	& * ( - gde3w(ji,jj,jk) + gde3w(ji,jj,jk-1) ) &
776	& + z_grav_10 * ( &
777	& ( zdrho_k (ji,jj,jk) - zdrho_k (ji,jj,jk-1) ) &
778	& * ( - gde3w(ji,jj,jk) + gde3w(ji,jj,jk-1) - z1_12 * ( zdz_k (ji,jj,jk) + zdz_k (ji,jj,jk-1) ) ) &
779	& - ( zdz_k (ji,jj,jk) - zdz_k (ji,jj,jk-1) ) &
780	& * ( rhd (ji,jj,jk) - rhd (ji,jj,jk-1) - z1_12 * ( zdrho_k(ji,jj,jk) + zdrho_k(ji,jj,jk-1) ) ) &
781	& )
782	END_3D
783
784	!----------------------------------------------------------------------------------------
785	! 5. compute and store elementary horizontal differences in provisional arrays
786	!----------------------------------------------------------------------------------------
787
788	DO_3D( 1, 0, 1, 0, 1, jpkm1 )
789	zdrhox(ji,jj,jk) = rhd (ji+1,jj ,jk) - rhd (ji,jj,jk )
790	zdzx (ji,jj,jk) = - gde3w(ji+1,jj ,jk) + gde3w(ji,jj,jk )
791	zdrhoy(ji,jj,jk) = rhd (ji ,jj+1,jk) - rhd (ji,jj,jk )
792	zdzy (ji,jj,jk) = - gde3w(ji ,jj+1,jk) + gde3w(ji,jj,jk )
793	END_3D
794
795	CALL lbc_lnk( 'dynhpg', zdrhox, 'U', 1., zdzx, 'U', 1., zdrhoy, 'V', 1., zdzy, 'V', 1. )
796
797	!-------------------------------------------------------------------------
798	! 6. compute harmonic averages using eq. 5.18
799	!-------------------------------------------------------------------------
800
801	DO_3D( 0, 1, 0, 1, 1, jpkm1 )
802	cffu = 2._wp * zdrhox(ji-1,jj ,jk) * zdrhox(ji,jj,jk )
803	IF( cffu > zep ) THEN
804	zdrho_i(ji,jj,jk) = cffu / ( zdrhox(ji-1,jj,jk) + zdrhox(ji,jj,jk) )
805	ELSE
806	zdrho_i(ji,jj,jk ) = 0._wp
807	ENDIF
808
809	cffx = 2._wp * zdzx (ji-1,jj ,jk) * zdzx (ji,jj,jk )
810	IF( cffx > zep ) THEN
811	zdz_i(ji,jj,jk) = cffx / ( zdzx(ji-1,jj,jk) + zdzx(ji,jj,jk) )
812	ELSE
813	zdz_i(ji,jj,jk) = 0._wp
814	ENDIF
815
816	cffv = 2._wp * zdrhoy(ji ,jj-1,jk) * zdrhoy(ji,jj,jk )
817	IF( cffv > zep ) THEN
818	zdrho_j(ji,jj,jk) = cffv / ( zdrhoy(ji,jj-1,jk) + zdrhoy(ji,jj,jk) )
819	ELSE
820	zdrho_j(ji,jj,jk) = 0._wp
821	ENDIF
822
823	cffy = 2._wp * zdzy (ji ,jj-1,jk) * zdzy (ji,jj,jk )
824	IF( cffy > zep ) THEN
825	zdz_j(ji,jj,jk) = cffy / ( zdzy(ji,jj-1,jk) + zdzy(ji,jj,jk) )
826	ELSE
827	zdz_j(ji,jj,jk) = 0._wp
828	ENDIF
829	END_3D
830
831	!!! Note that zdzx, zdzy, zdzz, zdrhox, zdrhoy and zdrhoz should NOT be used beyond this point
832
833	!----------------------------------------------------------------------------------
834	! 6B. apply boundary conditions at side boundaries using 5.36-5.37
835	!----------------------------------------------------------------------------------
836
837	DO jk = 1, jpkm1
838	zz_drho_i(:,:) = zdrho_i(:,:,jk)
839	zz_dz_i (:,:) = zdz_i (:,:,jk)
840	zz_drho_j(:,:) = zdrho_j(:,:,jk)
841	zz_dz_j (:,:) = zdz_j (:,:,jk)
842	DO_2D( 0, 1, 0, 1)
843	! Walls coming from left: should check from 2 to jpi-1 (and jpj=2-jpj)
844	IF (ji < jpi) THEN
845	IF ( umask(ji,jj,jk) > 0.5_wp .AND. umask(ji-1,jj,jk) < 0.5_wp .AND. umask(ji+1,jj,jk) > 0.5_wp) THEN
846	zz_drho_i(ji,jj) = aco_bc_hor * ( rhd (ji+1,jj,jk) - rhd (ji,jj,jk) ) - bco_bc_hor * zdrho_i(ji+1,jj,jk)
847	zz_dz_i (ji,jj) = aco_bc_hor * (-gde3w(ji+1,jj,jk) + gde3w(ji,jj,jk) ) - bco_bc_hor * zdz_i (ji+1,jj,jk)
848	END IF
849	END IF
850	! Walls coming from right: should check from 3 to jpi (and jpj=2-jpj)
851	IF (ji > 2) THEN
852	IF ( umask(ji,jj,jk) < 0.5_wp .AND. umask(ji-1,jj,jk) > 0.5_wp .AND. umask(ji-2,jj,jk) > 0.5_wp) THEN
853	zz_drho_i(ji,jj) = aco_bc_hor * ( rhd (ji,jj,jk) - rhd (ji-1,jj,jk) ) - bco_bc_hor * zdrho_i(ji-1,jj,jk)
854	zz_dz_i (ji,jj) = aco_bc_hor * (-gde3w(ji,jj,jk) + gde3w(ji-1,jj,jk) ) - bco_bc_hor * zdz_i (ji-1,jj,jk)
855	END IF
856	END IF
857	! Walls coming from left: should check from 2 to jpj-1 (and jpi=2-jpi)
858	IF (jj < jpj) THEN
859	IF ( vmask(ji,jj,jk) > 0.5_wp .AND. vmask(ji,jj-1,jk) < 0.5_wp .AND. vmask(ji,jj+1,jk) > 0.5_wp) THEN
860	zz_drho_j(ji,jj) = aco_bc_hor * ( rhd (ji,jj+1,jk) - rhd (ji,jj,jk) ) - bco_bc_hor * zdrho_j(ji,jj+1,jk)
861	zz_dz_j (ji,jj) = aco_bc_hor * (-gde3w(ji,jj+1,jk) + gde3w(ji,jj,jk) ) - bco_bc_hor * zdz_j (ji,jj+1,jk)
862	END IF
863	END IF
864	! Walls coming from right: should check from 3 to jpj (and jpi=2-jpi)
865	IF (jj > 2) THEN
866	IF ( vmask(ji,jj,jk) < 0.5_wp .AND. vmask(ji,jj-1,jk) > 0.5_wp .AND. vmask(ji,jj-2,jk) > 0.5_wp) THEN
867	zz_drho_j(ji,jj) = aco_bc_hor * ( rhd (ji,jj,jk) - rhd (ji,jj-1,jk) ) - bco_bc_hor * zdrho_j(ji,jj-1,jk)
868	zz_dz_j (ji,jj) = aco_bc_hor * (-gde3w(ji,jj,jk) + gde3w(ji,jj-1,jk) ) - bco_bc_hor * zdz_j (ji,jj-1,jk)
869	END IF
870	END IF
871	END_2D
872	zdrho_i(:,:,jk) = zz_drho_i(:,:)
873	zdz_i (:,:,jk) = zz_dz_i (:,:)
874	zdrho_j(:,:,jk) = zz_drho_j(:,:)
875	zdz_j (:,:,jk) = zz_dz_j (:,:)
876	END DO
877
878	!--------------------------------------------------------------
879	! 7. Calculate integrals on side faces
880	!-------------------------------------------------------------
881
882	DO_3D( 0, 0, 0, 0, 1, jpkm1 )
883	! two -ve signs cancel in next two lines (within zcoef0 and because gde3w is a depth not a height)
884	z_rho_i(ji,jj,jk) = zcoef0 * ( rhd (ji+1,jj,jk) + rhd (ji,jj,jk) ) &
885	& * ( gde3w(ji+1,jj,jk) - gde3w(ji,jj,jk) )
886	IF ( umask(ji-1, jj, jk) > 0.5 .OR. umask(ji+1, jj, jk) > 0.5 ) THEN
887	z_rho_i(ji,jj,jk) = z_rho_i(ji,jj,jk) - z_grav_10 * ( &
888	& ( zdrho_i (ji+1,jj,jk) - zdrho_i (ji,jj,jk) ) &
889	& * ( - gde3w(ji+1,jj,jk) + gde3w(ji,jj,jk) - z1_12 * ( zdz_i (ji+1,jj,jk) + zdz_i (ji,jj,jk) ) ) &
890	& - ( zdz_i (ji+1,jj,jk) - zdz_i (ji,jj,jk) ) &
891	& * ( rhd (ji+1,jj,jk) - rhd (ji,jj,jk) - z1_12 * ( zdrho_i(ji+1,jj,jk) + zdrho_i(ji,jj,jk) ) ) &
892	& )
893	END IF
894
895	z_rho_j(ji,jj,jk) = zcoef0 * ( rhd (ji,jj+1,jk) + rhd (ji,jj,jk) ) &
896	& * ( gde3w(ji,jj+1,jk) - gde3w(ji,jj,jk) )
897	IF ( vmask(ji, jj-1, jk) > 0.5 .OR. vmask(ji, jj+1, jk) > 0.5 ) THEN
898	z_rho_j(ji,jj,jk) = z_rho_j(ji,jj,jk) - z_grav_10 * ( &
899	& ( zdrho_j (ji,jj+1,jk) - zdrho_j (ji,jj,jk) ) &
900	& * ( - gde3w(ji,jj+1,jk) + gde3w(ji,jj,jk) - z1_12 * ( zdz_j (ji,jj+1,jk) + zdz_j (ji,jj,jk) ) ) &
901	& - ( zdz_j (ji,jj+1,jk) - zdz_j (ji,jj,jk) ) &
902	& * ( rhd (ji,jj+1,jk) - rhd (ji,jj,jk) - z1_12 * ( zdrho_j(ji,jj+1,jk) + zdrho_j(ji,jj,jk) ) ) &
903	& )
904	END IF
905	END_3D
906
907	!--------------------------------------------------------------
908	! 8. Integrate in the vertical
909	!-------------------------------------------------------------
910	!
911	! ---------------
912	! Surface value
913	! ---------------
914	DO_2D( 0, 0, 0, 0 )
915	zhpi(ji,jj,1) = ( z_rho_k(ji,jj,1) - z_rho_k(ji+1,jj ,1) - z_rho_i(ji,jj,1) ) * r1_e1u(ji,jj)
916	zhpj(ji,jj,1) = ( z_rho_k(ji,jj,1) - z_rho_k(ji ,jj+1,1) - z_rho_j(ji,jj,1) ) * r1_e2v(ji,jj)
917	IF( ln_wd_il ) THEN
918	zhpi(ji,jj,1) = zhpi(ji,jj,1) * zcpx(ji,jj)
919	zhpj(ji,jj,1) = zhpj(ji,jj,1) * zcpy(ji,jj)
920	ENDIF
921	! add to the general momentum trend
922	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + zhpi(ji,jj,1)
923	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + zhpj(ji,jj,1)
924	END_2D
925
926	! ----------------
927	! interior value (2=<jk=<jpkm1)
928	! ----------------
929	DO_3D( 0, 0, 0, 0, 2, jpkm1 )
930	! hydrostatic pressure gradient along s-surfaces
931	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
932	& + ( ( z_rho_k(ji,jj,jk) - z_rho_k(ji+1,jj,jk ) ) &
933	& - ( z_rho_i(ji,jj,jk) - z_rho_i(ji ,jj,jk-1) ) ) * r1_e1u(ji,jj)
934	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
935	& + ( ( z_rho_k(ji,jj,jk) - z_rho_k(ji,jj+1,jk ) ) &
936	& -( z_rho_j(ji,jj,jk) - z_rho_j(ji,jj ,jk-1) ) ) * r1_e2v(ji,jj)
937	IF( ln_wd_il ) THEN
938	zhpi(ji,jj,jk) = zhpi(ji,jj,jk) * zcpx(ji,jj)
939	zhpj(ji,jj,jk) = zhpj(ji,jj,jk) * zcpy(ji,jj)
940	ENDIF
941	! add to the general momentum trend
942	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + zhpi(ji,jj,jk)
943	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + zhpj(ji,jj,jk)
944	END_3D
945	!
946	IF( ln_wd_il ) DEALLOCATE( zcpx, zcpy )
947	!
948	END SUBROUTINE hpg_djc
949
950
951	SUBROUTINE hpg_prj( kt, Kmm, puu, pvv, Krhs )
952	!!---------------------------------------------------------------------
953	!! * ROUTINE hpg_prj *
954	!!
955	!! ** Method : s-coordinate case.
956	!! A Pressure-Jacobian horizontal pressure gradient method
957	!! based on the constrained cubic-spline interpolation for
958	!! all vertical coordinate systems
959	!!
960	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
961	!!----------------------------------------------------------------------
962	INTEGER, PARAMETER :: polynomial_type = 1 ! 1: cubic spline, 2: linear
963	INTEGER , INTENT( in ) :: kt ! ocean time-step index
964	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
965	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
966	!!
967	INTEGER :: ji, jj, jk, jkk ! dummy loop indices
968	REAL(wp) :: zcoef0, znad ! local scalars
969	!
970	!! The local variables for the correction term
971	INTEGER :: jk1, jis, jid, jjs, jjd
972	LOGICAL :: ll_tmp1, ll_tmp2 ! local logical variables
973	REAL(wp) :: zuijk, zvijk, zpwes, zpwed, zpnss, zpnsd, zdeps
974	REAL(wp) :: zrhdt1
975	REAL(wp) :: zdpdx1, zdpdx2, zdpdy1, zdpdy2
976	REAL(wp), DIMENSION(jpi,jpj) :: zpgu, zpgv ! 2D workspace
977	REAL(wp), DIMENSION(jpi,jpj) :: zsshu_n, zsshv_n
978	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zdept, zrhh
979	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zu, zv, fsp, xsp, asp, bsp, csp, dsp
980	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: zcpx, zcpy !W/D pressure filter
981	!!----------------------------------------------------------------------
982	!
983	IF( kt == nit000 ) THEN
984	IF(lwp) WRITE(numout,*)
985	IF(lwp) WRITE(numout,*) 'dyn:hpg_prj : hydrostatic pressure gradient trend'
986	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, cubic spline pressure Jacobian'
987	ENDIF
988
989	! Local constant initialization
990	zcoef0 = - grav
991	znad = 1._wp
992	IF( ln_linssh ) znad = 1._wp
993	!
994	! ---------------
995	! Surface pressure gradient to be removed
996	! ---------------
997	DO_2D( 0, 0, 0, 0 )
998	zpgu(ji,jj) = - grav * ( ssh(ji+1,jj,Kmm) - ssh(ji,jj,Kmm) ) * r1_e1u(ji,jj)
999	zpgv(ji,jj) = - grav * ( ssh(ji,jj+1,Kmm) - ssh(ji,jj,Kmm) ) * r1_e2v(ji,jj)
1000	END_2D
1001	!
1002	IF( ln_wd_il ) THEN
1003	ALLOCATE( zcpx(jpi,jpj) , zcpy(jpi,jpj) )
1004	DO_2D( 0, 0, 0, 0 )
1005	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
1006	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) .AND. &
1007	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) ) &
1008	& > rn_wdmin1 + rn_wdmin2
1009	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji+1,jj,Kmm) ) > 1.E-12 ) .AND. ( &
1010	& MAX( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
1011	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) + rn_wdmin1 + rn_wdmin2 )
1012
1013	IF(ll_tmp1) THEN
1014	zcpx(ji,jj) = 1.0_wp
1015	ELSE IF(ll_tmp2) THEN
1016	! no worries about ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm) = 0, it won't happen ! here
1017	zcpx(ji,jj) = ABS( (ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
1018	& / (ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm)) )
1019
1020	zcpx(ji,jj) = max(min( zcpx(ji,jj) , 1.0_wp),0.0_wp)
1021	ELSE
1022	zcpx(ji,jj) = 0._wp
1023	END IF
1024
1025	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
1026	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) .AND. &
1027	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) ) &
1028	& > rn_wdmin1 + rn_wdmin2
1029	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji,jj+1,Kmm) ) > 1.E-12 ) .AND. ( &
1030	& MAX( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
1031	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) + rn_wdmin1 + rn_wdmin2 )
1032
1033	IF(ll_tmp1) THEN
1034	zcpy(ji,jj) = 1.0_wp
1035	ELSE IF(ll_tmp2) THEN
1036	! no worries about ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm) = 0, it won't happen ! here
1037	zcpy(ji,jj) = ABS( (ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
1038	& / (ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm)) )
1039	zcpy(ji,jj) = max(min( zcpy(ji,jj) , 1.0_wp),0.0_wp)
1040
1041	ELSE
1042	zcpy(ji,jj) = 0._wp
1043	ENDIF
1044	END_2D
1045	CALL lbc_lnk( 'dynhpg', zcpx, 'U', 1.0_wp, zcpy, 'V', 1.0_wp )
1046	ENDIF
1047
1048	! Clean 3-D work arrays
1049	zhpi(:,:,:) = 0._wp
1050	zrhh(:,:,:) = rhd(:,:,:)
1051
1052	! Preparing vertical density profile "zrhh(:,:,:)" for hybrid-sco coordinate
1053	DO_2D( 1, 1, 1, 1 )
1054	jk = mbkt(ji,jj)
1055	IF( jk <= 1 ) THEN ; zrhh(ji,jj, : ) = 0._wp
1056	ELSEIF( jk == 2 ) THEN ; zrhh(ji,jj,jk+1:jpk) = rhd(ji,jj,jk)
1057	ELSEIF( jk < jpkm1 ) THEN
1058	DO jkk = jk+1, jpk
1059	zrhh(ji,jj,jkk) = interp1(gde3w(ji,jj,jkk ), gde3w(ji,jj,jkk-1), &
1060	& gde3w(ji,jj,jkk-2), zrhh (ji,jj,jkk-1), zrhh(ji,jj,jkk-2))
1061	END DO
1062	ENDIF
1063	END_2D
1064
1065	! Transfer the depth of "T(:,:,:)" to vertical coordinate "zdept(:,:,:)"
1066	DO_2D( 1, 1, 1, 1 )
1067	zdept(ji,jj,1) = 0.5_wp * e3w(ji,jj,1,Kmm) - ssh(ji,jj,Kmm)
1068	END_2D
1069
1070	DO_3D( 1, 1, 1, 1, 2, jpk )
1071	zdept(ji,jj,jk) = zdept(ji,jj,jk-1) + e3w(ji,jj,jk,Kmm)
1072	END_3D
1073
1074	fsp(:,:,:) = zrhh (:,:,:)
1075	xsp(:,:,:) = zdept(:,:,:)
1076
1077	! Construct the vertical density profile with the
1078	! constrained cubic spline interpolation
1079	! rho(z) = asp + bspz + cspz^2 + dsp*z^3
1080	CALL cspline( fsp, xsp, asp, bsp, csp, dsp, polynomial_type )
1081
1082	! Integrate the hydrostatic pressure "zhpi(:,:,:)" at "T(ji,jj,1)"
1083	DO_2D( 0, 1, 0, 1 )
1084	zrhdt1 = zrhh(ji,jj,1) - interp3( zdept(ji,jj,1), asp(ji,jj,1), bsp(ji,jj,1), &
1085	& csp(ji,jj,1), dsp(ji,jj,1) ) * 0.25_wp * e3w(ji,jj,1,Kmm)
1086
1087	! assuming linear profile across the top half surface layer
1088	zhpi(ji,jj,1) = 0.5_wp * e3w(ji,jj,1,Kmm) * zrhdt1
1089	END_2D
1090
1091	! Calculate the pressure "zhpi(:,:,:)" at "T(ji,jj,2:jpkm1)"
1092	DO_3D( 0, 1, 0, 1, 2, jpkm1 )
1093	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + &
1094	& integ_spline( zdept(ji,jj,jk-1), zdept(ji,jj,jk), &
1095	& asp (ji,jj,jk-1), bsp (ji,jj,jk-1), &
1096	& csp (ji,jj,jk-1), dsp (ji,jj,jk-1) )
1097	END_3D
1098
1099	! Z coordinate of U(ji,jj,1:jpkm1) and V(ji,jj,1:jpkm1)
1100
1101	! Prepare zsshu_n and zsshv_n
1102	DO_2D( 0, 0, 0, 0 )
1103	!!gm BUG ? if it is ssh at u- & v-point then it should be:
1104	! zsshu_n(ji,jj) = (e1e2t(ji,jj) * ssh(ji,jj,Kmm) + e1e2t(ji+1,jj) * ssh(ji+1,jj,Kmm)) * &
1105	! & r1_e1e2u(ji,jj) * umask(ji,jj,1) * 0.5_wp
1106	! zsshv_n(ji,jj) = (e1e2t(ji,jj) * ssh(ji,jj,Kmm) + e1e2t(ji,jj+1) * ssh(ji,jj+1,Kmm)) * &
1107	! & r1_e1e2v(ji,jj) * vmask(ji,jj,1) * 0.5_wp
1108	!!gm not this:
1109	zsshu_n(ji,jj) = (e1e2u(ji,jj) * ssh(ji,jj,Kmm) + e1e2u(ji+1, jj) * ssh(ji+1,jj,Kmm)) * &
1110	& r1_e1e2u(ji,jj) * umask(ji,jj,1) * 0.5_wp
1111	zsshv_n(ji,jj) = (e1e2v(ji,jj) * ssh(ji,jj,Kmm) + e1e2v(ji+1, jj) * ssh(ji,jj+1,Kmm)) * &
1112	& r1_e1e2v(ji,jj) * vmask(ji,jj,1) * 0.5_wp
1113	END_2D
1114
1115	CALL lbc_lnk ('dynhpg', zsshu_n, 'U', 1.0_wp, zsshv_n, 'V', 1.0_wp )
1116
1117	DO_2D( 0, 0, 0, 0 )
1118	zu(ji,jj,1) = - ( e3u(ji,jj,1,Kmm) - zsshu_n(ji,jj) )
1119	zv(ji,jj,1) = - ( e3v(ji,jj,1,Kmm) - zsshv_n(ji,jj) )
1120	END_2D
1121
1122	DO_3D( 0, 0, 0, 0, 2, jpkm1 )
1123	zu(ji,jj,jk) = zu(ji,jj,jk-1) - e3u(ji,jj,jk,Kmm)
1124	zv(ji,jj,jk) = zv(ji,jj,jk-1) - e3v(ji,jj,jk,Kmm)
1125	END_3D
1126
1127	DO_3D( 0, 0, 0, 0, 1, jpkm1 )
1128	zu(ji,jj,jk) = zu(ji,jj,jk) + 0.5_wp * e3u(ji,jj,jk,Kmm)
1129	zv(ji,jj,jk) = zv(ji,jj,jk) + 0.5_wp * e3v(ji,jj,jk,Kmm)
1130	END_3D
1131
1132	DO_3D( 0, 0, 0, 0, 1, jpkm1 )
1133	zu(ji,jj,jk) = MIN( zu(ji,jj,jk) , MAX( -zdept(ji,jj,jk) , -zdept(ji+1,jj,jk) ) )
1134	zu(ji,jj,jk) = MAX( zu(ji,jj,jk) , MIN( -zdept(ji,jj,jk) , -zdept(ji+1,jj,jk) ) )
1135	zv(ji,jj,jk) = MIN( zv(ji,jj,jk) , MAX( -zdept(ji,jj,jk) , -zdept(ji,jj+1,jk) ) )
1136	zv(ji,jj,jk) = MAX( zv(ji,jj,jk) , MIN( -zdept(ji,jj,jk) , -zdept(ji,jj+1,jk) ) )
1137	END_3D
1138
1139
1140	DO_3D( 0, 0, 0, 0, 1, jpkm1 )
1141	zpwes = 0._wp; zpwed = 0._wp
1142	zpnss = 0._wp; zpnsd = 0._wp
1143	zuijk = zu(ji,jj,jk)
1144	zvijk = zv(ji,jj,jk)
1145
1146	!!!!! for u equation
1147	IF( jk <= mbku(ji,jj) ) THEN
1148	IF( -zdept(ji+1,jj,jk) >= -zdept(ji,jj,jk) ) THEN
1149	jis = ji + 1; jid = ji
1150	ELSE
1151	jis = ji; jid = ji +1
1152	ENDIF
1153
1154	! integrate the pressure on the shallow side
1155	jk1 = jk
1156	DO WHILE ( -zdept(jis,jj,jk1) > zuijk )
1157	IF( jk1 == mbku(ji,jj) ) THEN
1158	zuijk = -zdept(jis,jj,jk1)
1159	EXIT
1160	ENDIF
1161	zdeps = MIN(zdept(jis,jj,jk1+1), -zuijk)
1162	zpwes = zpwes + &
1163	integ_spline(zdept(jis,jj,jk1), zdeps, &
1164	asp(jis,jj,jk1), bsp(jis,jj,jk1), &
1165	csp(jis,jj,jk1), dsp(jis,jj,jk1))
1166	jk1 = jk1 + 1
1167	END DO
1168
1169	! integrate the pressure on the deep side
1170	jk1 = jk
1171	DO WHILE ( -zdept(jid,jj,jk1) < zuijk )
1172	IF( jk1 == 1 ) THEN
1173	zdeps = zdept(jid,jj,1) + MIN(zuijk, ssh(jid,jj,Kmm)*znad)
1174	zrhdt1 = zrhh(jid,jj,1) - interp3(zdept(jid,jj,1), asp(jid,jj,1), &
1175	bsp(jid,jj,1), csp(jid,jj,1), &
1176	dsp(jid,jj,1)) * zdeps
1177	zpwed = zpwed + 0.5_wp * (zrhh(jid,jj,1) + zrhdt1) * zdeps
1178	EXIT
1179	ENDIF
1180	zdeps = MAX(zdept(jid,jj,jk1-1), -zuijk)
1181	zpwed = zpwed + &
1182	integ_spline(zdeps, zdept(jid,jj,jk1), &
1183	asp(jid,jj,jk1-1), bsp(jid,jj,jk1-1), &
1184	csp(jid,jj,jk1-1), dsp(jid,jj,jk1-1) )
1185	jk1 = jk1 - 1
1186	END DO
1187
1188	! update the momentum trends in u direction
1189
1190	zdpdx1 = zcoef0 * r1_e1u(ji,jj) * ( zhpi(ji+1,jj,jk) - zhpi(ji,jj,jk) )
1191	IF( .NOT.ln_linssh ) THEN
1192	zdpdx2 = zcoef0 * r1_e1u(ji,jj) * &
1193	& ( REAL(jis-jid, wp) * (zpwes + zpwed) + (ssh(ji+1,jj,Kmm)-ssh(ji,jj,Kmm)) )
1194	ELSE
1195	zdpdx2 = zcoef0 * r1_e1u(ji,jj) * REAL(jis-jid, wp) * (zpwes + zpwed)
1196	ENDIF
1197	IF( ln_wd_il ) THEN
1198	zdpdx1 = zdpdx1 * zcpx(ji,jj) * wdrampu(ji,jj)
1199	zdpdx2 = zdpdx2 * zcpx(ji,jj) * wdrampu(ji,jj)
1200	ENDIF
1201	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + (zdpdx1 + zdpdx2 - zpgu(ji,jj)) * umask(ji,jj,jk)
1202	ENDIF
1203
1204	!!!!! for v equation
1205	IF( jk <= mbkv(ji,jj) ) THEN
1206	IF( -zdept(ji,jj+1,jk) >= -zdept(ji,jj,jk) ) THEN
1207	jjs = jj + 1; jjd = jj
1208	ELSE
1209	jjs = jj ; jjd = jj + 1
1210	ENDIF
1211
1212	! integrate the pressure on the shallow side
1213	jk1 = jk
1214	DO WHILE ( -zdept(ji,jjs,jk1) > zvijk )
1215	IF( jk1 == mbkv(ji,jj) ) THEN
1216	zvijk = -zdept(ji,jjs,jk1)
1217	EXIT
1218	ENDIF
1219	zdeps = MIN(zdept(ji,jjs,jk1+1), -zvijk)
1220	zpnss = zpnss + &
1221	integ_spline(zdept(ji,jjs,jk1), zdeps, &
1222	asp(ji,jjs,jk1), bsp(ji,jjs,jk1), &
1223	csp(ji,jjs,jk1), dsp(ji,jjs,jk1) )
1224	jk1 = jk1 + 1
1225	END DO
1226
1227	! integrate the pressure on the deep side
1228	jk1 = jk
1229	DO WHILE ( -zdept(ji,jjd,jk1) < zvijk )
1230	IF( jk1 == 1 ) THEN
1231	zdeps = zdept(ji,jjd,1) + MIN(zvijk, ssh(ji,jjd,Kmm)*znad)
1232	zrhdt1 = zrhh(ji,jjd,1) - interp3(zdept(ji,jjd,1), asp(ji,jjd,1), &
1233	bsp(ji,jjd,1), csp(ji,jjd,1), &
1234	dsp(ji,jjd,1) ) * zdeps
1235	zpnsd = zpnsd + 0.5_wp * (zrhh(ji,jjd,1) + zrhdt1) * zdeps
1236	EXIT
1237	ENDIF
1238	zdeps = MAX(zdept(ji,jjd,jk1-1), -zvijk)
1239	zpnsd = zpnsd + &
1240	integ_spline(zdeps, zdept(ji,jjd,jk1), &
1241	asp(ji,jjd,jk1-1), bsp(ji,jjd,jk1-1), &
1242	csp(ji,jjd,jk1-1), dsp(ji,jjd,jk1-1) )
1243	jk1 = jk1 - 1
1244	END DO
1245
1246
1247	! update the momentum trends in v direction
1248
1249	zdpdy1 = zcoef0 * r1_e2v(ji,jj) * ( zhpi(ji,jj+1,jk) - zhpi(ji,jj,jk) )
1250	IF( .NOT.ln_linssh ) THEN
1251	zdpdy2 = zcoef0 * r1_e2v(ji,jj) * &
1252	( REAL(jjs-jjd, wp) * (zpnss + zpnsd) + (ssh(ji,jj+1,Kmm)-ssh(ji,jj,Kmm)) )
1253	ELSE
1254	zdpdy2 = zcoef0 * r1_e2v(ji,jj) * REAL(jjs-jjd, wp) * (zpnss + zpnsd )
1255	ENDIF
1256	IF( ln_wd_il ) THEN
1257	zdpdy1 = zdpdy1 * zcpy(ji,jj) * wdrampv(ji,jj)
1258	zdpdy2 = zdpdy2 * zcpy(ji,jj) * wdrampv(ji,jj)
1259	ENDIF
1260
1261	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + (zdpdy1 + zdpdy2 - zpgv(ji,jj)) * vmask(ji,jj,jk)
1262	ENDIF
1263	!
1264	END_3D
1265	!
1266	IF( ln_wd_il ) DEALLOCATE( zcpx, zcpy )
1267	!
1268	END SUBROUTINE hpg_prj
1269
1270
1271	SUBROUTINE cspline( fsp, xsp, asp, bsp, csp, dsp, polynomial_type )
1272	!!----------------------------------------------------------------------
1273	!! * ROUTINE cspline *
1274	!!
1275	!! ** Purpose : constrained cubic spline interpolation
1276	!!
1277	!! ** Method : f(x) = asp + bspx + cspx^2 + dsp*x^3
1278	!!
1279	!! Reference: CJC Kruger, Constrained Cubic Spline Interpoltation
1280	!!----------------------------------------------------------------------
1281	REAL(wp), DIMENSION(:,:,:), INTENT(in ) :: fsp, xsp ! value and coordinate
1282	REAL(wp), DIMENSION(:,:,:), INTENT( out) :: asp, bsp, csp, dsp ! coefficients of the interpoated function
1283	INTEGER , INTENT(in ) :: polynomial_type ! 1: cubic spline ; 2: Linear
1284	!
1285	INTEGER :: ji, jj, jk ! dummy loop indices
1286	INTEGER :: jpi, jpj, jpkm1
1287	REAL(wp) :: zdf1, zdf2, zddf1, zddf2, ztmp1, ztmp2, zdxtmp
1288	REAL(wp) :: zdxtmp1, zdxtmp2, zalpha
1289	REAL(wp) :: zdf(size(fsp,3))
1290	!!----------------------------------------------------------------------
1291	!
1292	!!gm WHAT !!!!! THIS IS VERY DANGEROUS !!!!!
1293	jpi = size(fsp,1)
1294	jpj = size(fsp,2)
1295	jpkm1 = MAX( 1, size(fsp,3) - 1 )
1296	!
1297	IF (polynomial_type == 1) THEN ! Constrained Cubic Spline
1298	DO ji = 1, jpi
1299	DO jj = 1, jpj
1300	!!Fritsch&Butland's method, 1984 (preferred, but more computation)
1301	! DO jk = 2, jpkm1-1
1302	! zdxtmp1 = xsp(ji,jj,jk) - xsp(ji,jj,jk-1)
1303	! zdxtmp2 = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1304	! zdf1 = ( fsp(ji,jj,jk) - fsp(ji,jj,jk-1) ) / zdxtmp1
1305	! zdf2 = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp2
1306	!
1307	! zalpha = ( zdxtmp1 + 2._wp * zdxtmp2 ) / ( zdxtmp1 + zdxtmp2 ) / 3._wp
1308	!
1309	! IF(zdf1 * zdf2 <= 0._wp) THEN
1310	! zdf(jk) = 0._wp
1311	! ELSE
1312	! zdf(jk) = zdf1 * zdf2 / ( ( 1._wp - zalpha ) * zdf1 + zalpha * zdf2 )
1313	! ENDIF
1314	! END DO
1315
1316	!!Simply geometric average
1317	DO jk = 2, jpkm1-1
1318	zdf1 = (fsp(ji,jj,jk ) - fsp(ji,jj,jk-1)) / (xsp(ji,jj,jk ) - xsp(ji,jj,jk-1))
1319	zdf2 = (fsp(ji,jj,jk+1) - fsp(ji,jj,jk )) / (xsp(ji,jj,jk+1) - xsp(ji,jj,jk ))
1320
1321	IF(zdf1 * zdf2 <= 0._wp) THEN
1322	zdf(jk) = 0._wp
1323	ELSE
1324	zdf(jk) = 2._wp * zdf1 * zdf2 / (zdf1 + zdf2)
1325	ENDIF
1326	END DO
1327
1328	zdf(1) = 1.5_wp * ( fsp(ji,jj,2) - fsp(ji,jj,1) ) / &
1329	& ( xsp(ji,jj,2) - xsp(ji,jj,1) ) - 0.5_wp * zdf(2)
1330	zdf(jpkm1) = 1.5_wp * ( fsp(ji,jj,jpkm1) - fsp(ji,jj,jpkm1-1) ) / &
1331	& ( xsp(ji,jj,jpkm1) - xsp(ji,jj,jpkm1-1) ) - 0.5_wp * zdf(jpkm1 - 1)
1332
1333	DO jk = 1, jpkm1 - 1
1334	zdxtmp = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1335	ztmp1 = (zdf(jk+1) + 2._wp * zdf(jk)) / zdxtmp
1336	ztmp2 = 6._wp * (fsp(ji,jj,jk+1) - fsp(ji,jj,jk)) / zdxtmp / zdxtmp
1337	zddf1 = -2._wp * ztmp1 + ztmp2
1338	ztmp1 = (2._wp * zdf(jk+1) + zdf(jk)) / zdxtmp
1339	zddf2 = 2._wp * ztmp1 - ztmp2
1340
1341	dsp(ji,jj,jk) = (zddf2 - zddf1) / 6._wp / zdxtmp
1342	csp(ji,jj,jk) = ( xsp(ji,jj,jk+1) * zddf1 - xsp(ji,jj,jk)*zddf2 ) / 2._wp / zdxtmp
1343	bsp(ji,jj,jk) = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp - &
1344	& csp(ji,jj,jk) * ( xsp(ji,jj,jk+1) + xsp(ji,jj,jk) ) - &
1345	& dsp(ji,jj,jk) * ((xsp(ji,jj,jk+1) + xsp(ji,jj,jk))**2 - &
1346	& xsp(ji,jj,jk+1) * xsp(ji,jj,jk))
1347	asp(ji,jj,jk) = fsp(ji,jj,jk) - xsp(ji,jj,jk) * (bsp(ji,jj,jk) + &
1348	& (xsp(ji,jj,jk) * (csp(ji,jj,jk) + &
1349	& dsp(ji,jj,jk) * xsp(ji,jj,jk))))
1350	END DO
1351	END DO
1352	END DO
1353
1354	ELSEIF ( polynomial_type == 2 ) THEN ! Linear
1355	DO ji = 1, jpi
1356	DO jj = 1, jpj
1357	DO jk = 1, jpkm1-1
1358	zdxtmp =xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1359	ztmp1 = fsp(ji,jj,jk+1) - fsp(ji,jj,jk)
1360
1361	dsp(ji,jj,jk) = 0._wp
1362	csp(ji,jj,jk) = 0._wp
1363	bsp(ji,jj,jk) = ztmp1 / zdxtmp
1364	asp(ji,jj,jk) = fsp(ji,jj,jk) - bsp(ji,jj,jk) * xsp(ji,jj,jk)
1365	END DO
1366	END DO
1367	END DO
1368	!
1369	ELSE
1370	CALL ctl_stop( 'invalid polynomial type in cspline' )
1371	ENDIF
1372	!
1373	END SUBROUTINE cspline
1374
1375
1376	FUNCTION interp1(x, xl, xr, fl, fr) RESULT(f)
1377	!!----------------------------------------------------------------------
1378	!! * ROUTINE interp1 *
1379	!!
1380	!! ** Purpose : 1-d linear interpolation
1381	!!
1382	!! ** Method : interpolation is straight forward
1383	!! extrapolation is also permitted (no value limit)
1384	!!----------------------------------------------------------------------
1385	REAL(wp), INTENT(in) :: x, xl, xr, fl, fr
1386	REAL(wp) :: f ! result of the interpolation (extrapolation)
1387	REAL(wp) :: zdeltx
1388	!!----------------------------------------------------------------------
1389	!
1390	zdeltx = xr - xl
1391	IF( abs(zdeltx) <= 10._wp * EPSILON(x) ) THEN
1392	f = 0.5_wp * (fl + fr)
1393	ELSE
1394	f = ( (x - xl ) * fr - ( x - xr ) * fl ) / zdeltx
1395	ENDIF
1396	!
1397	END FUNCTION interp1
1398
1399
1400	FUNCTION interp2( x, a, b, c, d ) RESULT(f)
1401	!!----------------------------------------------------------------------
1402	!! * ROUTINE interp1 *
1403	!!
1404	!! ** Purpose : 1-d constrained cubic spline interpolation
1405	!!
1406	!! ** Method : cubic spline interpolation
1407	!!
1408	!!----------------------------------------------------------------------
1409	REAL(wp), INTENT(in) :: x, a, b, c, d
1410	REAL(wp) :: f ! value from the interpolation
1411	!!----------------------------------------------------------------------
1412	!
1413	f = a + x* ( b + x * ( c + d * x ) )
1414	!
1415	END FUNCTION interp2
1416
1417
1418	FUNCTION interp3( x, a, b, c, d ) RESULT(f)
1419	!!----------------------------------------------------------------------
1420	!! * ROUTINE interp1 *
1421	!!
1422	!! ** Purpose : Calculate the first order of derivative of
1423	!! a cubic spline function y=a+bx+cx^2+d*x^3
1424	!!
1425	!! ** Method : f=dy/dx=b+2cx+3dx^2
1426	!!
1427	!!----------------------------------------------------------------------
1428	REAL(wp), INTENT(in) :: x, a, b, c, d
1429	REAL(wp) :: f ! value from the interpolation
1430	!!----------------------------------------------------------------------
1431	!
1432	f = b + x * ( 2._wp * c + 3._wp * d * x)
1433	!
1434	END FUNCTION interp3
1435
1436
1437	FUNCTION integ_spline( xl, xr, a, b, c, d ) RESULT(f)
1438	!!----------------------------------------------------------------------
1439	!! * ROUTINE interp1 *
1440	!!
1441	!! ** Purpose : 1-d constrained cubic spline integration
1442	!!
1443	!! ** Method : integrate polynomial a+bx+cx^2+dx^3 from xl to xr
1444	!!
1445	!!----------------------------------------------------------------------
1446	REAL(wp), INTENT(in) :: xl, xr, a, b, c, d
1447	REAL(wp) :: za1, za2, za3
1448	REAL(wp) :: f ! integration result
1449	!!----------------------------------------------------------------------
1450	!
1451	za1 = 0.5_wp * b
1452	za2 = c / 3.0_wp
1453	za3 = 0.25_wp * d
1454	!
1455	f = xr * ( a + xr * ( za1 + xr * ( za2 + za3 * xr ) ) ) - &
1456	& xl * ( a + xl * ( za1 + xl * ( za2 + za3 * xl ) ) )
1457	!
1458	END FUNCTION integ_spline
1459
1460	!!======================================================================
1461	END MODULE dynhpg

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