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dynhpg.F90 in NEMO/branches/2019/dev_r11943_MERGE_2019/src/OCE/DYN – NEMO

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source: NEMO/branches/2019/dev_r11943_MERGE_2019/src/OCE/DYN/dynhpg.F90 @ 12150

Last change on this file since 12150 was 12150, checked in by davestorkey, 4 years ago
2019/dev_r11943_MERGE_2019: Merge in UKMO_MERGE_2019.
Property svn:keywords set to `Id`
File size: 73.2 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	!!----------------------------------------------------------------------
20
21	!!----------------------------------------------------------------------
22	!! dyn_hpg : update the momentum trend with the now horizontal
23	!! gradient of the hydrostatic pressure
24	!! dyn_hpg_init : initialisation and control of options
25	!! hpg_zco : z-coordinate scheme
26	!! hpg_zps : z-coordinate plus partial steps (interpolation)
27	!! hpg_sco : s-coordinate (standard jacobian formulation)
28	!! hpg_isf : s-coordinate (sco formulation) adapted to ice shelf
29	!! hpg_djc : s-coordinate (Density Jacobian with Cubic polynomial)
30	!! hpg_prj : s-coordinate (Pressure Jacobian with Cubic polynomial)
31	!!----------------------------------------------------------------------
32	USE oce ! ocean dynamics and tracers
33	USE isf_oce , ONLY : risfload ! ice shelf (risfload variable)
34	USE isfload , ONLY : isf_load ! ice shelf (isf_load routine )
35	USE sbc_oce ! surface variable (only for the flag with ice shelf)
36	USE dom_oce ! ocean space and time domain
37	USE wet_dry ! wetting and drying
38	USE phycst ! physical constants
39	USE trd_oce ! trends: ocean variables
40	USE trddyn ! trend manager: dynamics
41	USE zpshde ! partial step: hor. derivative (zps_hde routine)
42	!
43	USE in_out_manager ! I/O manager
44	USE prtctl ! Print control
45	USE lbclnk ! lateral boundary condition
46	USE lib_mpp ! MPP library
47	USE eosbn2 ! compute density
48	USE timing ! Timing
49	USE iom
50
51	IMPLICIT NONE
52	PRIVATE
53
54	PUBLIC dyn_hpg ! routine called by step module
55	PUBLIC dyn_hpg_init ! routine called by opa module
56
57	! !!* Namelist namdyn_hpg : hydrostatic pressure gradient
58	LOGICAL, PUBLIC :: ln_hpg_zco !: z-coordinate - full steps
59	LOGICAL, PUBLIC :: ln_hpg_zps !: z-coordinate - partial steps (interpolation)
60	LOGICAL, PUBLIC :: ln_hpg_sco !: s-coordinate (standard jacobian formulation)
61	LOGICAL, PUBLIC :: ln_hpg_djc !: s-coordinate (Density Jacobian with Cubic polynomial)
62	LOGICAL, PUBLIC :: ln_hpg_prj !: s-coordinate (Pressure Jacobian scheme)
63	LOGICAL, PUBLIC :: ln_hpg_isf !: s-coordinate similar to sco modify for isf
64
65	! !! Flag to control the type of hydrostatic pressure gradient
66	INTEGER, PARAMETER :: np_ERROR =-10 ! error in specification of lateral diffusion
67	INTEGER, PARAMETER :: np_zco = 0 ! z-coordinate - full steps
68	INTEGER, PARAMETER :: np_zps = 1 ! z-coordinate - partial steps (interpolation)
69	INTEGER, PARAMETER :: np_sco = 2 ! s-coordinate (standard jacobian formulation)
70	INTEGER, PARAMETER :: np_djc = 3 ! s-coordinate (Density Jacobian with Cubic polynomial)
71	INTEGER, PARAMETER :: np_prj = 4 ! s-coordinate (Pressure Jacobian scheme)
72	INTEGER, PARAMETER :: np_isf = 5 ! s-coordinate similar to sco modify for isf
73	!
74	INTEGER, PUBLIC :: nhpg !: type of pressure gradient scheme used ! (deduced from ln_hpg_... flags) (PUBLIC for TAM)
75
76	!! * Substitutions
77	# include "vectopt_loop_substitute.h90"
78	!!----------------------------------------------------------------------
79	!! NEMO/OCE 4.0 , NEMO Consortium (2018)
80	!! $Id$
81	!! Software governed by the CeCILL license (see ./LICENSE)
82	!!----------------------------------------------------------------------
83	CONTAINS
84
85	SUBROUTINE dyn_hpg( kt, Kmm, puu, pvv, Krhs )
86	!!---------------------------------------------------------------------
87	!! * ROUTINE dyn_hpg *
88	!!
89	!! ** Method : Call the hydrostatic pressure gradient routine
90	!! using the scheme defined in the namelist
91	!!
92	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
93	!! - send trends to trd_dyn for futher diagnostics (l_trddyn=T)
94	!!----------------------------------------------------------------------
95	INTEGER , INTENT( in ) :: kt ! ocean time-step index
96	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
97	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
98	!
99	REAL(wp), ALLOCATABLE, DIMENSION(:,:,:) :: ztrdu, ztrdv
100	!!----------------------------------------------------------------------
101	!
102	IF( ln_timing ) CALL timing_start('dyn_hpg')
103	!
104	IF( l_trddyn ) THEN ! Temporary saving of puu(:,:,:,Krhs) and pvv(:,:,:,Krhs) trends (l_trddyn)
105	ALLOCATE( ztrdu(jpi,jpj,jpk) , ztrdv(jpi,jpj,jpk) )
106	ztrdu(:,:,:) = puu(:,:,:,Krhs)
107	ztrdv(:,:,:) = pvv(:,:,:,Krhs)
108	ENDIF
109	!
110	SELECT CASE ( nhpg ) ! Hydrostatic pressure gradient computation
111	CASE ( np_zco ) ; CALL hpg_zco ( kt, Kmm, puu, pvv, Krhs ) ! z-coordinate
112	CASE ( np_zps ) ; CALL hpg_zps ( kt, Kmm, puu, pvv, Krhs ) ! z-coordinate plus partial steps (interpolation)
113	CASE ( np_sco ) ; CALL hpg_sco ( kt, Kmm, puu, pvv, Krhs ) ! s-coordinate (standard jacobian formulation)
114	CASE ( np_djc ) ; CALL hpg_djc ( kt, Kmm, puu, pvv, Krhs ) ! s-coordinate (Density Jacobian with Cubic polynomial)
115	CASE ( np_prj ) ; CALL hpg_prj ( kt, Kmm, puu, pvv, Krhs ) ! s-coordinate (Pressure Jacobian scheme)
116	CASE ( np_isf ) ; CALL hpg_isf ( kt, Kmm, puu, pvv, Krhs ) ! s-coordinate similar to sco modify for ice shelf
117	END SELECT
118	!
119	IF( l_trddyn ) THEN ! save the hydrostatic pressure gradient trends for momentum trend diagnostics
120	ztrdu(:,:,:) = puu(:,:,:,Krhs) - ztrdu(:,:,:)
121	ztrdv(:,:,:) = pvv(:,:,:,Krhs) - ztrdv(:,:,:)
122	CALL trd_dyn( ztrdu, ztrdv, jpdyn_hpg, kt, Kmm )
123	DEALLOCATE( ztrdu , ztrdv )
124	ENDIF
125	!
126	IF(ln_ctl) CALL prt_ctl( tab3d_1=puu(:,:,:,Krhs), clinfo1=' hpg - Ua: ', mask1=umask, &
127	& tab3d_2=pvv(:,:,:,Krhs), clinfo2= ' Va: ', mask2=vmask, clinfo3='dyn' )
128	!
129	IF( ln_timing ) CALL timing_stop('dyn_hpg')
130	!
131	END SUBROUTINE dyn_hpg
132
133
134	SUBROUTINE dyn_hpg_init( Kmm )
135	!!----------------------------------------------------------------------
136	!! * ROUTINE dyn_hpg_init *
137	!!
138	!! ** Purpose : initializations for the hydrostatic pressure gradient
139	!! computation and consistency control
140	!!
141	!! ** Action : Read the namelist namdyn_hpg and check the consistency
142	!! with the type of vertical coordinate used (zco, zps, sco)
143	!!----------------------------------------------------------------------
144	INTEGER, INTENT( in ) :: Kmm ! ocean time level index
145	!
146	INTEGER :: ioptio = 0 ! temporary integer
147	INTEGER :: ios ! Local integer output status for namelist read
148	!!
149	INTEGER :: ji, jj, jk, ikt ! dummy loop indices ISF
150	REAL(wp) :: znad
151	REAL(wp), ALLOCATABLE, DIMENSION(:,:,:) :: zts_top, zrhd ! hypothesys on isf density
152	REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: zrhdtop_isf ! density at bottom of ISF
153	REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: ziceload ! density at bottom of ISF
154	!!
155	NAMELIST/namdyn_hpg/ ln_hpg_zco, ln_hpg_zps, ln_hpg_sco, &
156	& ln_hpg_djc, ln_hpg_prj, ln_hpg_isf
157	!!----------------------------------------------------------------------
158	!
159	READ ( numnam_ref, namdyn_hpg, IOSTAT = ios, ERR = 901)
160	901 IF( ios /= 0 ) CALL ctl_nam ( ios , 'namdyn_hpg in reference namelist' )
161	!
162	READ ( numnam_cfg, namdyn_hpg, IOSTAT = ios, ERR = 902 )
163	902 IF( ios > 0 ) CALL ctl_nam ( ios , 'namdyn_hpg in configuration namelist' )
164	IF(lwm) WRITE ( numond, namdyn_hpg )
165	!
166	IF(lwp) THEN ! Control print
167	WRITE(numout,*)
168	WRITE(numout,*) 'dyn_hpg_init : hydrostatic pressure gradient initialisation'
169	WRITE(numout,*) '~~~~~~~~~~~~'
170	WRITE(numout,*) ' Namelist namdyn_hpg : choice of hpg scheme'
171	WRITE(numout,*) ' z-coord. - full steps ln_hpg_zco = ', ln_hpg_zco
172	WRITE(numout,*) ' z-coord. - partial steps (interpolation) ln_hpg_zps = ', ln_hpg_zps
173	WRITE(numout,*) ' s-coord. (standard jacobian formulation) ln_hpg_sco = ', ln_hpg_sco
174	WRITE(numout,*) ' s-coord. (standard jacobian formulation) for isf ln_hpg_isf = ', ln_hpg_isf
175	WRITE(numout,*) ' s-coord. (Density Jacobian: Cubic polynomial) ln_hpg_djc = ', ln_hpg_djc
176	WRITE(numout,*) ' s-coord. (Pressure Jacobian: Cubic polynomial) ln_hpg_prj = ', ln_hpg_prj
177	ENDIF
178	!
179	IF( ln_hpg_djc ) &
180	& CALL ctl_stop('dyn_hpg_init : Density Jacobian: Cubic polynominal method', &
181	& ' currently disabled (bugs under investigation).' , &
182	& ' Please select either ln_hpg_sco or ln_hpg_prj instead' )
183	!
184	IF( .NOT.ln_linssh .AND. .NOT.(ln_hpg_sco.OR.ln_hpg_prj.OR.ln_hpg_isf) ) &
185	& CALL ctl_stop('dyn_hpg_init : non-linear free surface requires either ', &
186	& ' the standard jacobian formulation hpg_sco or ' , &
187	& ' the pressure jacobian formulation hpg_prj' )
188	!
189	IF( ln_hpg_isf ) THEN
190	IF( .NOT. ln_isfcav ) CALL ctl_stop( ' hpg_isf not available if ln_isfcav = false ' )
191	ELSE
192	IF( ln_isfcav ) CALL ctl_stop( 'Only hpg_isf has been corrected to work with ice shelf cavity.' )
193	ENDIF
194	!
195	! ! Set nhpg from ln_hpg_... flags & consistency check
196	nhpg = np_ERROR
197	ioptio = 0
198	IF( ln_hpg_zco ) THEN ; nhpg = np_zco ; ioptio = ioptio +1 ; ENDIF
199	IF( ln_hpg_zps ) THEN ; nhpg = np_zps ; ioptio = ioptio +1 ; ENDIF
200	IF( ln_hpg_sco ) THEN ; nhpg = np_sco ; ioptio = ioptio +1 ; ENDIF
201	IF( ln_hpg_djc ) THEN ; nhpg = np_djc ; ioptio = ioptio +1 ; ENDIF
202	IF( ln_hpg_prj ) THEN ; nhpg = np_prj ; ioptio = ioptio +1 ; ENDIF
203	IF( ln_hpg_isf ) THEN ; nhpg = np_isf ; ioptio = ioptio +1 ; ENDIF
204	!
205	IF( ioptio /= 1 ) CALL ctl_stop( 'NO or several hydrostatic pressure gradient options used' )
206	!
207	IF(lwp) THEN
208	WRITE(numout,*)
209	SELECT CASE( nhpg )
210	CASE( np_zco ) ; WRITE(numout,*) ' ==>>> z-coord. - full steps '
211	CASE( np_zps ) ; WRITE(numout,*) ' ==>>> z-coord. - partial steps (interpolation)'
212	CASE( np_sco ) ; WRITE(numout,*) ' ==>>> s-coord. (standard jacobian formulation)'
213	CASE( np_djc ) ; WRITE(numout,*) ' ==>>> s-coord. (Density Jacobian: Cubic polynomial)'
214	CASE( np_prj ) ; WRITE(numout,*) ' ==>>> s-coord. (Pressure Jacobian: Cubic polynomial)'
215	CASE( np_isf ) ; WRITE(numout,*) ' ==>>> s-coord. (standard jacobian formulation) for isf'
216	END SELECT
217	WRITE(numout,*)
218	ENDIF
219	!
220	END SUBROUTINE dyn_hpg_init
221
222
223	SUBROUTINE hpg_zco( kt, Kmm, puu, pvv, Krhs )
224	!!---------------------------------------------------------------------
225	!! * ROUTINE hpg_zco *
226	!!
227	!! ** Method : z-coordinate case, levels are horizontal surfaces.
228	!! The now hydrostatic pressure gradient at a given level, jk,
229	!! is computed by taking the vertical integral of the in-situ
230	!! density gradient along the model level from the suface to that
231	!! level: zhpi = grav .....
232	!! zhpj = grav .....
233	!! add it to the general momentum trend (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)).
234	!! puu(:,:,:,Krhs) = puu(:,:,:,Krhs) - 1/e1u * zhpi
235	!! pvv(:,:,:,Krhs) = pvv(:,:,:,Krhs) - 1/e2v * zhpj
236	!!
237	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
238	!!----------------------------------------------------------------------
239	INTEGER , INTENT( in ) :: kt ! ocean time-step index
240	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
241	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
242	!
243	INTEGER :: ji, jj, jk ! dummy loop indices
244	REAL(wp) :: zcoef0, zcoef1 ! temporary scalars
245	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zhpj
246	!!----------------------------------------------------------------------
247	!
248	IF( kt == nit000 ) THEN
249	IF(lwp) WRITE(numout,*)
250	IF(lwp) WRITE(numout,*) 'dyn:hpg_zco : hydrostatic pressure gradient trend'
251	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ z-coordinate case '
252	ENDIF
253
254	zcoef0 = - grav * 0.5_wp ! Local constant initialization
255
256	! Surface value
257	DO jj = 2, jpjm1
258	DO ji = fs_2, fs_jpim1 ! vector opt.
259	zcoef1 = zcoef0 * e3w(ji,jj,1,Kmm)
260	! hydrostatic pressure gradient
261	zhpi(ji,jj,1) = zcoef1 * ( rhd(ji+1,jj,1) - rhd(ji,jj,1) ) * r1_e1u(ji,jj)
262	zhpj(ji,jj,1) = zcoef1 * ( rhd(ji,jj+1,1) - rhd(ji,jj,1) ) * r1_e2v(ji,jj)
263	! add to the general momentum trend
264	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + zhpi(ji,jj,1)
265	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + zhpj(ji,jj,1)
266	END DO
267	END DO
268
269	!
270	! interior value (2=<jk=<jpkm1)
271	DO jk = 2, jpkm1
272	DO jj = 2, jpjm1
273	DO ji = fs_2, fs_jpim1 ! vector opt.
274	zcoef1 = zcoef0 * e3w(ji,jj,jk,Kmm)
275	! hydrostatic pressure gradient
276	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
277	& + zcoef1 * ( ( rhd(ji+1,jj,jk)+rhd(ji+1,jj,jk-1) ) &
278	& - ( rhd(ji ,jj,jk)+rhd(ji ,jj,jk-1) ) ) * r1_e1u(ji,jj)
279
280	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
281	& + zcoef1 * ( ( rhd(ji,jj+1,jk)+rhd(ji,jj+1,jk-1) ) &
282	& - ( rhd(ji,jj, jk)+rhd(ji,jj ,jk-1) ) ) * r1_e2v(ji,jj)
283	! add to the general momentum trend
284	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + zhpi(ji,jj,jk)
285	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + zhpj(ji,jj,jk)
286	END DO
287	END DO
288	END DO
289	!
290	END SUBROUTINE hpg_zco
291
292
293	SUBROUTINE hpg_zps( kt, Kmm, puu, pvv, Krhs )
294	!!---------------------------------------------------------------------
295	!! * ROUTINE hpg_zps *
296	!!
297	!! ** Method : z-coordinate plus partial steps case. blahblah...
298	!!
299	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
300	!!----------------------------------------------------------------------
301	INTEGER , INTENT( in ) :: kt ! ocean time-step index
302	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
303	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
304	!!
305	INTEGER :: ji, jj, jk ! dummy loop indices
306	INTEGER :: iku, ikv ! temporary integers
307	REAL(wp) :: zcoef0, zcoef1, zcoef2, zcoef3 ! temporary scalars
308	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zhpj
309	REAL(wp), DIMENSION(jpi,jpj) :: zgtsu, zgtsv, zgru, zgrv
310	!!----------------------------------------------------------------------
311	!
312	IF( kt == nit000 ) THEN
313	IF(lwp) WRITE(numout,*)
314	IF(lwp) WRITE(numout,*) 'dyn:hpg_zps : hydrostatic pressure gradient trend'
315	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ z-coordinate with partial steps - vector optimization'
316	ENDIF
317
318	! Partial steps: Compute NOW horizontal gradient of t, s, rd at the last ocean level
319	CALL zps_hde( kt, Kmm, jpts, ts(:,:,:,:,Kmm), zgtsu, zgtsv, rhd, zgru , zgrv )
320
321	! Local constant initialization
322	zcoef0 = - grav * 0.5_wp
323
324	! Surface value (also valid in partial step case)
325	DO jj = 2, jpjm1
326	DO ji = fs_2, fs_jpim1 ! vector opt.
327	zcoef1 = zcoef0 * e3w(ji,jj,1,Kmm)
328	! hydrostatic pressure gradient
329	zhpi(ji,jj,1) = zcoef1 * ( rhd(ji+1,jj ,1) - rhd(ji,jj,1) ) * r1_e1u(ji,jj)
330	zhpj(ji,jj,1) = zcoef1 * ( rhd(ji ,jj+1,1) - rhd(ji,jj,1) ) * r1_e2v(ji,jj)
331	! add to the general momentum trend
332	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + zhpi(ji,jj,1)
333	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + zhpj(ji,jj,1)
334	END DO
335	END DO
336
337	! interior value (2=<jk=<jpkm1)
338	DO jk = 2, jpkm1
339	DO jj = 2, jpjm1
340	DO ji = fs_2, fs_jpim1 ! vector opt.
341	zcoef1 = zcoef0 * e3w(ji,jj,jk,Kmm)
342	! hydrostatic pressure gradient
343	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
344	& + zcoef1 * ( ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) ) &
345	& - ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) ) ) * r1_e1u(ji,jj)
346
347	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
348	& + zcoef1 * ( ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) ) &
349	& - ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) ) ) * r1_e2v(ji,jj)
350	! add to the general momentum trend
351	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + zhpi(ji,jj,jk)
352	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + zhpj(ji,jj,jk)
353	END DO
354	END DO
355	END DO
356
357	! partial steps correction at the last level (use zgru & zgrv computed in zpshde.F90)
358	DO jj = 2, jpjm1
359	DO ji = 2, jpim1
360	iku = mbku(ji,jj)
361	ikv = mbkv(ji,jj)
362	zcoef2 = zcoef0 * MIN( e3w(ji,jj,iku,Kmm), e3w(ji+1,jj ,iku,Kmm) )
363	zcoef3 = zcoef0 * MIN( e3w(ji,jj,ikv,Kmm), e3w(ji ,jj+1,ikv,Kmm) )
364	IF( iku > 1 ) THEN ! on i-direction (level 2 or more)
365	puu (ji,jj,iku,Krhs) = puu(ji,jj,iku,Krhs) - zhpi(ji,jj,iku) ! subtract old value
366	zhpi(ji,jj,iku) = zhpi(ji,jj,iku-1) & ! compute the new one
367	& + zcoef2 * ( rhd(ji+1,jj,iku-1) - rhd(ji,jj,iku-1) + zgru(ji,jj) ) * r1_e1u(ji,jj)
368	puu (ji,jj,iku,Krhs) = puu(ji,jj,iku,Krhs) + zhpi(ji,jj,iku) ! add the new one to the general momentum trend
369	ENDIF
370	IF( ikv > 1 ) THEN ! on j-direction (level 2 or more)
371	pvv (ji,jj,ikv,Krhs) = pvv(ji,jj,ikv,Krhs) - zhpj(ji,jj,ikv) ! subtract old value
372	zhpj(ji,jj,ikv) = zhpj(ji,jj,ikv-1) & ! compute the new one
373	& + zcoef3 * ( rhd(ji,jj+1,ikv-1) - rhd(ji,jj,ikv-1) + zgrv(ji,jj) ) * r1_e2v(ji,jj)
374	pvv (ji,jj,ikv,Krhs) = pvv(ji,jj,ikv,Krhs) + zhpj(ji,jj,ikv) ! add the new one to the general momentum trend
375	ENDIF
376	END DO
377	END DO
378	!
379	END SUBROUTINE hpg_zps
380
381
382	SUBROUTINE hpg_sco( kt, Kmm, puu, pvv, Krhs )
383	!!---------------------------------------------------------------------
384	!! * ROUTINE hpg_sco *
385	!!
386	!! ** Method : s-coordinate case. Jacobian scheme.
387	!! The now hydrostatic pressure gradient at a given level, jk,
388	!! is computed by taking the vertical integral of the in-situ
389	!! density gradient along the model level from the suface to that
390	!! level. s-coordinates (ln_sco): a corrective term is added
391	!! to the horizontal pressure gradient :
392	!! zhpi = grav ..... + 1/e1u mi(rhd) di[ grav dep3w ]
393	!! zhpj = grav ..... + 1/e2v mj(rhd) dj[ grav dep3w ]
394	!! add it to the general momentum trend (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)).
395	!! puu(:,:,:,Krhs) = puu(:,:,:,Krhs) - 1/e1u * zhpi
396	!! pvv(:,:,:,Krhs) = pvv(:,:,:,Krhs) - 1/e2v * zhpj
397	!!
398	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
399	!!----------------------------------------------------------------------
400	INTEGER , INTENT( in ) :: kt ! ocean time-step index
401	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
402	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
403	!!
404	INTEGER :: ji, jj, jk, jii, jjj ! dummy loop indices
405	REAL(wp) :: zcoef0, zuap, zvap, znad, ztmp ! temporary scalars
406	LOGICAL :: ll_tmp1, ll_tmp2 ! local logical variables
407	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zhpj
408	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: zcpx, zcpy !W/D pressure filter
409	!!----------------------------------------------------------------------
410	!
411	IF( ln_wd_il ) ALLOCATE(zcpx(jpi,jpj), zcpy(jpi,jpj))
412	!
413	IF( kt == nit000 ) THEN
414	IF(lwp) WRITE(numout,*)
415	IF(lwp) WRITE(numout,*) 'dyn:hpg_sco : hydrostatic pressure gradient trend'
416	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, OPA original scheme used'
417	ENDIF
418	!
419	zcoef0 = - grav * 0.5_wp
420	IF ( ln_linssh ) THEN ; znad = 0._wp ! Fixed volume: density anomaly
421	ELSE ; znad = 1._wp ! Variable volume: density
422	ENDIF
423	!
424	IF( ln_wd_il ) THEN
425	DO jj = 2, jpjm1
426	DO ji = 2, jpim1
427	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
428	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) .AND. &
429	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) ) &
430	& > rn_wdmin1 + rn_wdmin2
431	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji+1,jj,Kmm) ) > 1.E-12 ) .AND. ( &
432	& MAX( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
433	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) + rn_wdmin1 + rn_wdmin2 )
434
435	IF(ll_tmp1) THEN
436	zcpx(ji,jj) = 1.0_wp
437	ELSE IF(ll_tmp2) THEN
438	! no worries about ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm) = 0, it won't happen ! here
439	zcpx(ji,jj) = ABS( (ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
440	& / (ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm)) )
441	ELSE
442	zcpx(ji,jj) = 0._wp
443	END IF
444
445	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
446	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) .AND. &
447	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) ) &
448	& > rn_wdmin1 + rn_wdmin2
449	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji,jj+1,Kmm) ) > 1.E-12 ) .AND. ( &
450	& MAX( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
451	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) + rn_wdmin1 + rn_wdmin2 )
452
453	IF(ll_tmp1) THEN
454	zcpy(ji,jj) = 1.0_wp
455	ELSE IF(ll_tmp2) THEN
456	! no worries about ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm) = 0, it won't happen ! here
457	zcpy(ji,jj) = ABS( (ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
458	& / (ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm)) )
459	ELSE
460	zcpy(ji,jj) = 0._wp
461	END IF
462	END DO
463	END DO
464	CALL lbc_lnk_multi( 'dynhpg', zcpx, 'U', 1., zcpy, 'V', 1. )
465	END IF
466
467	! Surface value
468	DO jj = 2, jpjm1
469	DO ji = fs_2, fs_jpim1 ! vector opt.
470	! hydrostatic pressure gradient along s-surfaces
471	zhpi(ji,jj,1) = zcoef0 * ( e3w(ji+1,jj ,1,Kmm) * ( znad + rhd(ji+1,jj ,1) ) &
472	& - e3w(ji ,jj ,1,Kmm) * ( znad + rhd(ji ,jj ,1) ) ) * r1_e1u(ji,jj)
473	zhpj(ji,jj,1) = zcoef0 * ( e3w(ji ,jj+1,1,Kmm) * ( znad + rhd(ji ,jj+1,1) ) &
474	& - e3w(ji ,jj ,1,Kmm) * ( znad + rhd(ji ,jj ,1) ) ) * r1_e2v(ji,jj)
475	! s-coordinate pressure gradient correction
476	zuap = -zcoef0 * ( rhd (ji+1,jj,1) + rhd (ji,jj,1) + 2._wp * znad ) &
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) + 2._wp * znad ) &
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	!
488	! add to the general momentum trend
489	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + zhpi(ji,jj,1) + zuap
490	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + zhpj(ji,jj,1) + zvap
491	END DO
492	END DO
493
494	! interior value (2=<jk=<jpkm1)
495	DO jk = 2, jpkm1
496	DO jj = 2, jpjm1
497	DO ji = fs_2, fs_jpim1 ! vector opt.
498	! hydrostatic pressure gradient along s-surfaces
499	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + zcoef0 * r1_e1u(ji,jj) &
500	& * ( e3w(ji+1,jj,jk,Kmm) * ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) + 2*znad ) &
501	& - e3w(ji ,jj,jk,Kmm) * ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) + 2*znad ) )
502	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) + zcoef0 * r1_e2v(ji,jj) &
503	& * ( e3w(ji,jj+1,jk,Kmm) * ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) + 2*znad ) &
504	& - e3w(ji,jj ,jk,Kmm) * ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) + 2*znad ) )
505	! s-coordinate pressure gradient correction
506	zuap = -zcoef0 * ( rhd (ji+1,jj ,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
507	& * ( gde3w(ji+1,jj ,jk) - gde3w(ji,jj,jk) ) * r1_e1u(ji,jj)
508	zvap = -zcoef0 * ( rhd (ji ,jj+1,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
509	& * ( gde3w(ji ,jj+1,jk) - gde3w(ji,jj,jk) ) * r1_e2v(ji,jj)
510	!
511	IF( ln_wd_il ) THEN
512	zhpi(ji,jj,jk) = zhpi(ji,jj,jk) * zcpx(ji,jj)
513	zhpj(ji,jj,jk) = zhpj(ji,jj,jk) * zcpy(ji,jj)
514	zuap = zuap * zcpx(ji,jj)
515	zvap = zvap * zcpy(ji,jj)
516	ENDIF
517	!
518	! add to the general momentum trend
519	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + zhpi(ji,jj,jk) + zuap
520	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + zhpj(ji,jj,jk) + zvap
521	END DO
522	END DO
523	END DO
524	!
525	IF( ln_wd_il ) DEALLOCATE( zcpx , zcpy )
526	!
527	END SUBROUTINE hpg_sco
528
529
530	SUBROUTINE hpg_isf( kt, Kmm, puu, pvv, Krhs )
531	!!---------------------------------------------------------------------
532	!! * ROUTINE hpg_isf *
533	!!
534	!! ** Method : s-coordinate case. Jacobian scheme.
535	!! The now hydrostatic pressure gradient at a given level, jk,
536	!! is computed by taking the vertical integral of the in-situ
537	!! density gradient along the model level from the suface to that
538	!! level. s-coordinates (ln_sco): a corrective term is added
539	!! to the horizontal pressure gradient :
540	!! zhpi = grav ..... + 1/e1u mi(rhd) di[ grav dep3w ]
541	!! zhpj = grav ..... + 1/e2v mj(rhd) dj[ grav dep3w ]
542	!! add it to the general momentum trend (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)).
543	!! puu(:,:,:,Krhs) = puu(:,:,:,Krhs) - 1/e1u * zhpi
544	!! pvv(:,:,:,Krhs) = pvv(:,:,:,Krhs) - 1/e2v * zhpj
545	!! iceload is added
546	!!
547	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
548	!!----------------------------------------------------------------------
549	INTEGER , INTENT( in ) :: kt ! ocean time-step index
550	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
551	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
552	!!
553	INTEGER :: ji, jj, jk, ikt, iktp1i, iktp1j ! dummy loop indices
554	REAL(wp) :: zcoef0, zuap, zvap, znad ! temporary scalars
555	REAL(wp), DIMENSION(jpi,jpj,jpk ) :: zhpi, zhpj
556	REAL(wp), DIMENSION(jpi,jpj,jpts) :: zts_top
557	REAL(wp), DIMENSION(jpi,jpj) :: zrhdtop_oce
558	!!----------------------------------------------------------------------
559	!
560	zcoef0 = - grav * 0.5_wp ! Local constant initialization
561	!
562	znad=1._wp ! To use density and not density anomaly
563	!
564	! ! iniitialised to 0. zhpi zhpi
565	zhpi(:,:,:) = 0._wp ; zhpj(:,:,:) = 0._wp
566
567	! compute rhd at the ice/oce interface (ocean side)
568	! usefull to reduce residual current in the test case ISOMIP with no melting
569	DO ji = 1, jpi
570	DO jj = 1, jpj
571	ikt = mikt(ji,jj)
572	zts_top(ji,jj,1) = ts(ji,jj,ikt,1,Kmm)
573	zts_top(ji,jj,2) = ts(ji,jj,ikt,2,Kmm)
574	END DO
575	END DO
576	CALL eos( zts_top, risfdep, zrhdtop_oce )
577
578	!==================================================================================
579	!===== Compute surface value =====================================================
580	!==================================================================================
581	DO jj = 2, jpjm1
582	DO ji = fs_2, fs_jpim1 ! vector opt.
583	ikt = mikt(ji,jj)
584	iktp1i = mikt(ji+1,jj)
585	iktp1j = mikt(ji,jj+1)
586	! hydrostatic pressure gradient along s-surfaces and ice shelf pressure
587	! we assume ISF is in isostatic equilibrium
588	zhpi(ji,jj,1) = zcoef0 / e1u(ji,jj) * ( 0.5_wp * e3w(ji+1,jj,iktp1i,Kmm) &
589	& * ( 2._wp * znad + rhd(ji+1,jj,iktp1i) + zrhdtop_oce(ji+1,jj) ) &
590	& - 0.5_wp * e3w(ji,jj,ikt,Kmm) &
591	& * ( 2._wp * znad + rhd(ji,jj,ikt) + zrhdtop_oce(ji,jj) ) &
592	& + ( risfload(ji+1,jj) - risfload(ji,jj)) )
593	zhpj(ji,jj,1) = zcoef0 / e2v(ji,jj) * ( 0.5_wp * e3w(ji,jj+1,iktp1j,Kmm) &
594	& * ( 2._wp * znad + rhd(ji,jj+1,iktp1j) + zrhdtop_oce(ji,jj+1) ) &
595	& - 0.5_wp * e3w(ji,jj,ikt,Kmm) &
596	& * ( 2._wp * znad + rhd(ji,jj,ikt) + zrhdtop_oce(ji,jj) ) &
597	& + ( risfload(ji,jj+1) - risfload(ji,jj)) )
598	! s-coordinate pressure gradient correction (=0 if z coordinate)
599	zuap = -zcoef0 * ( rhd (ji+1,jj,1) + rhd (ji,jj,1) + 2._wp * znad ) &
600	& * ( gde3w(ji+1,jj,1) - gde3w(ji,jj,1) ) * r1_e1u(ji,jj)
601	zvap = -zcoef0 * ( rhd (ji,jj+1,1) + rhd (ji,jj,1) + 2._wp * znad ) &
602	& * ( gde3w(ji,jj+1,1) - gde3w(ji,jj,1) ) * r1_e2v(ji,jj)
603	! add to the general momentum trend
604	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + (zhpi(ji,jj,1) + zuap) * umask(ji,jj,1)
605	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + (zhpj(ji,jj,1) + zvap) * vmask(ji,jj,1)
606	END DO
607	END DO
608	!==================================================================================
609	!===== Compute interior value =====================================================
610	!==================================================================================
611	! interior value (2=<jk=<jpkm1)
612	DO jk = 2, jpkm1
613	DO jj = 2, jpjm1
614	DO ji = fs_2, fs_jpim1 ! vector opt.
615	! hydrostatic pressure gradient along s-surfaces
616	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + zcoef0 / e1u(ji,jj) &
617	& * ( e3w(ji+1,jj,jk,Kmm) * ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) + 2znad ) wmask(ji+1,jj,jk) &
618	& - e3w(ji ,jj,jk,Kmm) * ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) + 2znad ) wmask(ji ,jj,jk) )
619	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) + zcoef0 / e2v(ji,jj) &
620	& * ( e3w(ji,jj+1,jk,Kmm) * ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) + 2znad ) wmask(ji,jj+1,jk) &
621	& - e3w(ji,jj ,jk,Kmm) * ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) + 2znad ) wmask(ji,jj ,jk) )
622	! s-coordinate pressure gradient correction
623	zuap = -zcoef0 * ( rhd (ji+1,jj ,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
624	& * ( gde3w(ji+1,jj ,jk) - gde3w(ji,jj,jk) ) / e1u(ji,jj)
625	zvap = -zcoef0 * ( rhd (ji ,jj+1,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
626	& * ( gde3w(ji ,jj+1,jk) - gde3w(ji,jj,jk) ) / e2v(ji,jj)
627	! add to the general momentum trend
628	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + (zhpi(ji,jj,jk) + zuap) * umask(ji,jj,jk)
629	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + (zhpj(ji,jj,jk) + zvap) * vmask(ji,jj,jk)
630	END DO
631	END DO
632	END DO
633	!
634	END SUBROUTINE hpg_isf
635
636
637	SUBROUTINE hpg_djc( kt, Kmm, puu, pvv, Krhs )
638	!!---------------------------------------------------------------------
639	!! * ROUTINE hpg_djc *
640	!!
641	!! ** Method : Density Jacobian with Cubic polynomial scheme
642	!!
643	!! Reference: Shchepetkin and McWilliams, J. Geophys. Res., 108(C3), 3090, 2003
644	!!----------------------------------------------------------------------
645	INTEGER , INTENT( in ) :: kt ! ocean time-step index
646	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
647	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
648	!!
649	INTEGER :: ji, jj, jk ! dummy loop indices
650	REAL(wp) :: zcoef0, zep, cffw ! temporary scalars
651	REAL(wp) :: z1_10, cffu, cffx ! " "
652	REAL(wp) :: z1_12, cffv, cffy ! " "
653	LOGICAL :: ll_tmp1, ll_tmp2 ! local logical variables
654	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zhpj
655	REAL(wp), DIMENSION(jpi,jpj,jpk) :: dzx, dzy, dzz, dzu, dzv, dzw
656	REAL(wp), DIMENSION(jpi,jpj,jpk) :: drhox, drhoy, drhoz, drhou, drhov, drhow
657	REAL(wp), DIMENSION(jpi,jpj,jpk) :: rho_i, rho_j, rho_k
658	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: zcpx, zcpy !W/D pressure filter
659	!!----------------------------------------------------------------------
660	!
661	IF( ln_wd_il ) THEN
662	ALLOCATE( zcpx(jpi,jpj) , zcpy(jpi,jpj) )
663	DO jj = 2, jpjm1
664	DO ji = 2, jpim1
665	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
666	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) .AND. &
667	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) ) &
668	& > rn_wdmin1 + rn_wdmin2
669	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji+1,jj,Kmm) ) > 1.E-12 ) .AND. ( &
670	& MAX( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
671	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) + rn_wdmin1 + rn_wdmin2 )
672	IF(ll_tmp1) THEN
673	zcpx(ji,jj) = 1.0_wp
674	ELSE IF(ll_tmp2) THEN
675	! no worries about ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm) = 0, it won't happen ! here
676	zcpx(ji,jj) = ABS( (ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
677	& / (ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm)) )
678	ELSE
679	zcpx(ji,jj) = 0._wp
680	END IF
681
682	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
683	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) .AND. &
684	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) ) &
685	& > rn_wdmin1 + rn_wdmin2
686	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji,jj+1,Kmm) ) > 1.E-12 ) .AND. ( &
687	& MAX( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
688	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) + rn_wdmin1 + rn_wdmin2 )
689
690	IF(ll_tmp1) THEN
691	zcpy(ji,jj) = 1.0_wp
692	ELSE IF(ll_tmp2) THEN
693	! no worries about ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm) = 0, it won't happen ! here
694	zcpy(ji,jj) = ABS( (ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
695	& / (ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm)) )
696	ELSE
697	zcpy(ji,jj) = 0._wp
698	END IF
699	END DO
700	END DO
701	CALL lbc_lnk_multi( 'dynhpg', zcpx, 'U', 1., zcpy, 'V', 1. )
702	END IF
703
704	IF( kt == nit000 ) THEN
705	IF(lwp) WRITE(numout,*)
706	IF(lwp) WRITE(numout,*) 'dyn:hpg_djc : hydrostatic pressure gradient trend'
707	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, density Jacobian with cubic polynomial scheme'
708	ENDIF
709
710	! Local constant initialization
711	zcoef0 = - grav * 0.5_wp
712	z1_10 = 1._wp / 10._wp
713	z1_12 = 1._wp / 12._wp
714
715	!----------------------------------------------------------------------------------------
716	! compute and store in provisional arrays elementary vertical and horizontal differences
717	!----------------------------------------------------------------------------------------
718
719	!!bug gm Not a true bug, but... dzz=e3w for dzx, dzy verify what it is really
720
721	DO jk = 2, jpkm1
722	DO jj = 2, jpjm1
723	DO ji = fs_2, fs_jpim1 ! vector opt.
724	drhoz(ji,jj,jk) = rhd (ji ,jj ,jk) - rhd (ji,jj,jk-1)
725	dzz (ji,jj,jk) = gde3w(ji ,jj ,jk) - gde3w(ji,jj,jk-1)
726	drhox(ji,jj,jk) = rhd (ji+1,jj ,jk) - rhd (ji,jj,jk )
727	dzx (ji,jj,jk) = gde3w(ji+1,jj ,jk) - gde3w(ji,jj,jk )
728	drhoy(ji,jj,jk) = rhd (ji ,jj+1,jk) - rhd (ji,jj,jk )
729	dzy (ji,jj,jk) = gde3w(ji ,jj+1,jk) - gde3w(ji,jj,jk )
730	END DO
731	END DO
732	END DO
733
734	!-------------------------------------------------------------------------
735	! compute harmonic averages using eq. 5.18
736	!-------------------------------------------------------------------------
737	zep = 1.e-15
738
739	!!bug gm drhoz not defined at level 1 and used (jk-1 with jk=2)
740	!!bug gm idem for drhox, drhoy et ji=jpi and jj=jpj
741
742	DO jk = 2, jpkm1
743	DO jj = 2, jpjm1
744	DO ji = fs_2, fs_jpim1 ! vector opt.
745	cffw = 2._wp * drhoz(ji ,jj ,jk) * drhoz(ji,jj,jk-1)
746
747	cffu = 2._wp * drhox(ji+1,jj ,jk) * drhox(ji,jj,jk )
748	cffx = 2._wp * dzx (ji+1,jj ,jk) * dzx (ji,jj,jk )
749
750	cffv = 2._wp * drhoy(ji ,jj+1,jk) * drhoy(ji,jj,jk )
751	cffy = 2._wp * dzy (ji ,jj+1,jk) * dzy (ji,jj,jk )
752
753	IF( cffw > zep) THEN
754	drhow(ji,jj,jk) = 2._wp * drhoz(ji,jj,jk) * drhoz(ji,jj,jk-1) &
755	& / ( drhoz(ji,jj,jk) + drhoz(ji,jj,jk-1) )
756	ELSE
757	drhow(ji,jj,jk) = 0._wp
758	ENDIF
759
760	dzw(ji,jj,jk) = 2._wp * dzz(ji,jj,jk) * dzz(ji,jj,jk-1) &
761	& / ( dzz(ji,jj,jk) + dzz(ji,jj,jk-1) )
762
763	IF( cffu > zep ) THEN
764	drhou(ji,jj,jk) = 2._wp * drhox(ji+1,jj,jk) * drhox(ji,jj,jk) &
765	& / ( drhox(ji+1,jj,jk) + drhox(ji,jj,jk) )
766	ELSE
767	drhou(ji,jj,jk ) = 0._wp
768	ENDIF
769
770	IF( cffx > zep ) THEN
771	dzu(ji,jj,jk) = 2._wp * dzx(ji+1,jj,jk) * dzx(ji,jj,jk) &
772	& / ( dzx(ji+1,jj,jk) + dzx(ji,jj,jk) )
773	ELSE
774	dzu(ji,jj,jk) = 0._wp
775	ENDIF
776
777	IF( cffv > zep ) THEN
778	drhov(ji,jj,jk) = 2._wp * drhoy(ji,jj+1,jk) * drhoy(ji,jj,jk) &
779	& / ( drhoy(ji,jj+1,jk) + drhoy(ji,jj,jk) )
780	ELSE
781	drhov(ji,jj,jk) = 0._wp
782	ENDIF
783
784	IF( cffy > zep ) THEN
785	dzv(ji,jj,jk) = 2._wp * dzy(ji,jj+1,jk) * dzy(ji,jj,jk) &
786	& / ( dzy(ji,jj+1,jk) + dzy(ji,jj,jk) )
787	ELSE
788	dzv(ji,jj,jk) = 0._wp
789	ENDIF
790
791	END DO
792	END DO
793	END DO
794
795	!----------------------------------------------------------------------------------
796	! apply boundary conditions at top and bottom using 5.36-5.37
797	!----------------------------------------------------------------------------------
798	drhow(:,:, 1 ) = 1.5_wp * ( drhoz(:,:, 2 ) - drhoz(:,:, 1 ) ) - 0.5_wp * drhow(:,:, 2 )
799	drhou(:,:, 1 ) = 1.5_wp * ( drhox(:,:, 2 ) - drhox(:,:, 1 ) ) - 0.5_wp * drhou(:,:, 2 )
800	drhov(:,:, 1 ) = 1.5_wp * ( drhoy(:,:, 2 ) - drhoy(:,:, 1 ) ) - 0.5_wp * drhov(:,:, 2 )
801
802	drhow(:,:,jpk) = 1.5_wp * ( drhoz(:,:,jpk) - drhoz(:,:,jpkm1) ) - 0.5_wp * drhow(:,:,jpkm1)
803	drhou(:,:,jpk) = 1.5_wp * ( drhox(:,:,jpk) - drhox(:,:,jpkm1) ) - 0.5_wp * drhou(:,:,jpkm1)
804	drhov(:,:,jpk) = 1.5_wp * ( drhoy(:,:,jpk) - drhoy(:,:,jpkm1) ) - 0.5_wp * drhov(:,:,jpkm1)
805
806
807	!--------------------------------------------------------------
808	! Upper half of top-most grid box, compute and store
809	!-------------------------------------------------------------
810
811	!!bug gm : e3w-gde3w = 0.5*e3w .... and gde3w(2)-gde3w(1)=e3w(2) .... to be verified
812	! true if gde3w is really defined as the sum of the e3w scale factors as, it seems to me, it should be
813
814	DO jj = 2, jpjm1
815	DO ji = fs_2, fs_jpim1 ! vector opt.
816	rho_k(ji,jj,1) = -grav * ( e3w(ji,jj,1,Kmm) - gde3w(ji,jj,1) ) &
817	& * ( rhd(ji,jj,1) &
818	& + 0.5_wp * ( rhd (ji,jj,2) - rhd (ji,jj,1) ) &
819	& * ( e3w (ji,jj,1,Kmm) - gde3w(ji,jj,1) ) &
820	& / ( gde3w(ji,jj,2) - gde3w(ji,jj,1) ) )
821	END DO
822	END DO
823
824	!!bug gm : here also, simplification is possible
825	!!bug gm : optimisation: 1/10 and 1/12 the division should be done before the loop
826
827	DO jk = 2, jpkm1
828	DO jj = 2, jpjm1
829	DO ji = fs_2, fs_jpim1 ! vector opt.
830
831	rho_k(ji,jj,jk) = zcoef0 * ( rhd (ji,jj,jk) + rhd (ji,jj,jk-1) ) &
832	& * ( gde3w(ji,jj,jk) - gde3w(ji,jj,jk-1) ) &
833	& - grav * z1_10 * ( &
834	& ( drhow (ji,jj,jk) - drhow (ji,jj,jk-1) ) &
835	& * ( gde3w(ji,jj,jk) - gde3w(ji,jj,jk-1) - z1_12 * ( dzw (ji,jj,jk) + dzw (ji,jj,jk-1) ) ) &
836	& - ( dzw (ji,jj,jk) - dzw (ji,jj,jk-1) ) &
837	& * ( rhd (ji,jj,jk) - rhd (ji,jj,jk-1) - z1_12 * ( drhow(ji,jj,jk) + drhow(ji,jj,jk-1) ) ) &
838	& )
839
840	rho_i(ji,jj,jk) = zcoef0 * ( rhd (ji+1,jj,jk) + rhd (ji,jj,jk) ) &
841	& * ( gde3w(ji+1,jj,jk) - gde3w(ji,jj,jk) ) &
842	& - grav* z1_10 * ( &
843	& ( drhou (ji+1,jj,jk) - drhou (ji,jj,jk) ) &
844	& * ( gde3w(ji+1,jj,jk) - gde3w(ji,jj,jk) - z1_12 * ( dzu (ji+1,jj,jk) + dzu (ji,jj,jk) ) ) &
845	& - ( dzu (ji+1,jj,jk) - dzu (ji,jj,jk) ) &
846	& * ( rhd (ji+1,jj,jk) - rhd (ji,jj,jk) - z1_12 * ( drhou(ji+1,jj,jk) + drhou(ji,jj,jk) ) ) &
847	& )
848
849	rho_j(ji,jj,jk) = zcoef0 * ( rhd (ji,jj+1,jk) + rhd (ji,jj,jk) ) &
850	& * ( gde3w(ji,jj+1,jk) - gde3w(ji,jj,jk) ) &
851	& - grav* z1_10 * ( &
852	& ( drhov (ji,jj+1,jk) - drhov (ji,jj,jk) ) &
853	& * ( gde3w(ji,jj+1,jk) - gde3w(ji,jj,jk) - z1_12 * ( dzv (ji,jj+1,jk) + dzv (ji,jj,jk) ) ) &
854	& - ( dzv (ji,jj+1,jk) - dzv (ji,jj,jk) ) &
855	& * ( rhd (ji,jj+1,jk) - rhd (ji,jj,jk) - z1_12 * ( drhov(ji,jj+1,jk) + drhov(ji,jj,jk) ) ) &
856	& )
857
858	END DO
859	END DO
860	END DO
861	CALL lbc_lnk_multi( 'dynhpg', rho_k, 'W', 1., rho_i, 'U', 1., rho_j, 'V', 1. )
862
863	! ---------------
864	! Surface value
865	! ---------------
866	DO jj = 2, jpjm1
867	DO ji = fs_2, fs_jpim1 ! vector opt.
868	zhpi(ji,jj,1) = ( rho_k(ji+1,jj ,1) - rho_k(ji,jj,1) - rho_i(ji,jj,1) ) * r1_e1u(ji,jj)
869	zhpj(ji,jj,1) = ( rho_k(ji ,jj+1,1) - rho_k(ji,jj,1) - rho_j(ji,jj,1) ) * r1_e2v(ji,jj)
870	IF( ln_wd_il ) THEN
871	zhpi(ji,jj,1) = zhpi(ji,jj,1) * zcpx(ji,jj)
872	zhpj(ji,jj,1) = zhpj(ji,jj,1) * zcpy(ji,jj)
873	ENDIF
874	! add to the general momentum trend
875	puu(ji,jj,1,Krhs) = puu(ji,jj,1,Krhs) + zhpi(ji,jj,1)
876	pvv(ji,jj,1,Krhs) = pvv(ji,jj,1,Krhs) + zhpj(ji,jj,1)
877	END DO
878	END DO
879
880	! ----------------
881	! interior value (2=<jk=<jpkm1)
882	! ----------------
883	DO jk = 2, jpkm1
884	DO jj = 2, jpjm1
885	DO ji = fs_2, fs_jpim1 ! vector opt.
886	! hydrostatic pressure gradient along s-surfaces
887	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
888	& + ( ( rho_k(ji+1,jj,jk) - rho_k(ji,jj,jk ) ) &
889	& - ( rho_i(ji ,jj,jk) - rho_i(ji,jj,jk-1) ) ) * r1_e1u(ji,jj)
890	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
891	& + ( ( rho_k(ji,jj+1,jk) - rho_k(ji,jj,jk ) ) &
892	& -( rho_j(ji,jj ,jk) - rho_j(ji,jj,jk-1) ) ) * r1_e2v(ji,jj)
893	IF( ln_wd_il ) THEN
894	zhpi(ji,jj,jk) = zhpi(ji,jj,jk) * zcpx(ji,jj)
895	zhpj(ji,jj,jk) = zhpj(ji,jj,jk) * zcpy(ji,jj)
896	ENDIF
897	! add to the general momentum trend
898	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + zhpi(ji,jj,jk)
899	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + zhpj(ji,jj,jk)
900	END DO
901	END DO
902	END DO
903	!
904	IF( ln_wd_il ) DEALLOCATE( zcpx, zcpy )
905	!
906	END SUBROUTINE hpg_djc
907
908
909	SUBROUTINE hpg_prj( kt, Kmm, puu, pvv, Krhs )
910	!!---------------------------------------------------------------------
911	!! * ROUTINE hpg_prj *
912	!!
913	!! ** Method : s-coordinate case.
914	!! A Pressure-Jacobian horizontal pressure gradient method
915	!! based on the constrained cubic-spline interpolation for
916	!! all vertical coordinate systems
917	!!
918	!! ** Action : - Update (puu(:,:,:,Krhs),pvv(:,:,:,Krhs)) with the now hydrastatic pressure trend
919	!!----------------------------------------------------------------------
920	INTEGER, PARAMETER :: polynomial_type = 1 ! 1: cubic spline, 2: linear
921	INTEGER , INTENT( in ) :: kt ! ocean time-step index
922	INTEGER , INTENT( in ) :: Kmm, Krhs ! ocean time level indices
923	REAL(wp), DIMENSION(jpi,jpj,jpk,jpt), INTENT(inout) :: puu, pvv ! ocean velocities and RHS of momentum equation
924	!!
925	INTEGER :: ji, jj, jk, jkk ! dummy loop indices
926	REAL(wp) :: zcoef0, znad ! local scalars
927	!
928	!! The local variables for the correction term
929	INTEGER :: jk1, jis, jid, jjs, jjd
930	LOGICAL :: ll_tmp1, ll_tmp2 ! local logical variables
931	REAL(wp) :: zuijk, zvijk, zpwes, zpwed, zpnss, zpnsd, zdeps
932	REAL(wp) :: zrhdt1
933	REAL(wp) :: zdpdx1, zdpdx2, zdpdy1, zdpdy2
934	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zdept, zrhh
935	REAL(wp), DIMENSION(jpi,jpj,jpk) :: zhpi, zu, zv, fsp, xsp, asp, bsp, csp, dsp
936	REAL(wp), DIMENSION(jpi,jpj) :: zsshu_n, zsshv_n
937	REAL(wp), DIMENSION(:,:), ALLOCATABLE :: zcpx, zcpy !W/D pressure filter
938	!!----------------------------------------------------------------------
939	!
940	IF( kt == nit000 ) THEN
941	IF(lwp) WRITE(numout,*)
942	IF(lwp) WRITE(numout,*) 'dyn:hpg_prj : hydrostatic pressure gradient trend'
943	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, cubic spline pressure Jacobian'
944	ENDIF
945
946	! Local constant initialization
947	zcoef0 = - grav
948	znad = 1._wp
949	IF( ln_linssh ) znad = 0._wp
950
951	IF( ln_wd_il ) THEN
952	ALLOCATE( zcpx(jpi,jpj) , zcpy(jpi,jpj) )
953	DO jj = 2, jpjm1
954	DO ji = 2, jpim1
955	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
956	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) .AND. &
957	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) ) &
958	& > rn_wdmin1 + rn_wdmin2
959	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji+1,jj,Kmm) ) > 1.E-12 ) .AND. ( &
960	& MAX( ssh(ji,jj,Kmm) , ssh(ji+1,jj,Kmm) ) > &
961	& MAX( -ht_0(ji,jj) , -ht_0(ji+1,jj) ) + rn_wdmin1 + rn_wdmin2 )
962
963	IF(ll_tmp1) THEN
964	zcpx(ji,jj) = 1.0_wp
965	ELSE IF(ll_tmp2) THEN
966	! no worries about ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm) = 0, it won't happen ! here
967	zcpx(ji,jj) = ABS( (ssh(ji+1,jj,Kmm) + ht_0(ji+1,jj) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
968	& / (ssh(ji+1,jj,Kmm) - ssh(ji ,jj,Kmm)) )
969
970	zcpx(ji,jj) = max(min( zcpx(ji,jj) , 1.0_wp),0.0_wp)
971	ELSE
972	zcpx(ji,jj) = 0._wp
973	END IF
974
975	ll_tmp1 = MIN( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
976	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) .AND. &
977	& MAX( ssh(ji,jj,Kmm) + ht_0(ji,jj), ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) ) &
978	& > rn_wdmin1 + rn_wdmin2
979	ll_tmp2 = ( ABS( ssh(ji,jj,Kmm) - ssh(ji,jj+1,Kmm) ) > 1.E-12 ) .AND. ( &
980	& MAX( ssh(ji,jj,Kmm) , ssh(ji,jj+1,Kmm) ) > &
981	& MAX( -ht_0(ji,jj) , -ht_0(ji,jj+1) ) + rn_wdmin1 + rn_wdmin2 )
982
983	IF(ll_tmp1) THEN
984	zcpy(ji,jj) = 1.0_wp
985	ELSE IF(ll_tmp2) THEN
986	! no worries about ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm) = 0, it won't happen ! here
987	zcpy(ji,jj) = ABS( (ssh(ji,jj+1,Kmm) + ht_0(ji,jj+1) - ssh(ji,jj,Kmm) - ht_0(ji,jj)) &
988	& / (ssh(ji,jj+1,Kmm) - ssh(ji,jj ,Kmm)) )
989	zcpy(ji,jj) = max(min( zcpy(ji,jj) , 1.0_wp),0.0_wp)
990
991	ELSE
992	zcpy(ji,jj) = 0._wp
993	ENDIF
994	END DO
995	END DO
996	CALL lbc_lnk_multi( 'dynhpg', zcpx, 'U', 1., zcpy, 'V', 1. )
997	ENDIF
998
999	! Clean 3-D work arrays
1000	zhpi(:,:,:) = 0._wp
1001	zrhh(:,:,:) = rhd(:,:,:)
1002
1003	! Preparing vertical density profile "zrhh(:,:,:)" for hybrid-sco coordinate
1004	DO jj = 1, jpj
1005	DO ji = 1, jpi
1006	jk = mbkt(ji,jj)+1
1007	IF( jk <= 0 ) THEN ; zrhh(ji,jj, : ) = 0._wp
1008	ELSEIF( jk == 1 ) THEN ; zrhh(ji,jj,jk+1:jpk) = rhd(ji,jj,jk)
1009	ELSEIF( jk < jpkm1 ) THEN
1010	DO jkk = jk+1, jpk
1011	zrhh(ji,jj,jkk) = interp1(gde3w(ji,jj,jkk ), gde3w(ji,jj,jkk-1), &
1012	& gde3w(ji,jj,jkk-2), rhd (ji,jj,jkk-1), rhd(ji,jj,jkk-2))
1013	END DO
1014	ENDIF
1015	END DO
1016	END DO
1017
1018	! Transfer the depth of "T(:,:,:)" to vertical coordinate "zdept(:,:,:)"
1019	DO jj = 1, jpj
1020	DO ji = 1, jpi
1021	zdept(ji,jj,1) = 0.5_wp * e3w(ji,jj,1,Kmm) - ssh(ji,jj,Kmm) * znad
1022	END DO
1023	END DO
1024
1025	DO jk = 2, jpk
1026	DO jj = 1, jpj
1027	DO ji = 1, jpi
1028	zdept(ji,jj,jk) = zdept(ji,jj,jk-1) + e3w(ji,jj,jk,Kmm)
1029	END DO
1030	END DO
1031	END DO
1032
1033	fsp(:,:,:) = zrhh (:,:,:)
1034	xsp(:,:,:) = zdept(:,:,:)
1035
1036	! Construct the vertical density profile with the
1037	! constrained cubic spline interpolation
1038	! rho(z) = asp + bspz + cspz^2 + dsp*z^3
1039	CALL cspline( fsp, xsp, asp, bsp, csp, dsp, polynomial_type )
1040
1041	! Integrate the hydrostatic pressure "zhpi(:,:,:)" at "T(ji,jj,1)"
1042	DO jj = 2, jpj
1043	DO ji = 2, jpi
1044	zrhdt1 = zrhh(ji,jj,1) - interp3( zdept(ji,jj,1), asp(ji,jj,1), bsp(ji,jj,1), &
1045	& csp(ji,jj,1), dsp(ji,jj,1) ) * 0.25_wp * e3w(ji,jj,1,Kmm)
1046
1047	! assuming linear profile across the top half surface layer
1048	zhpi(ji,jj,1) = 0.5_wp * e3w(ji,jj,1,Kmm) * zrhdt1
1049	END DO
1050	END DO
1051
1052	! Calculate the pressure "zhpi(:,:,:)" at "T(ji,jj,2:jpkm1)"
1053	DO jk = 2, jpkm1
1054	DO jj = 2, jpj
1055	DO ji = 2, jpi
1056	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + &
1057	& integ_spline( zdept(ji,jj,jk-1), zdept(ji,jj,jk), &
1058	& asp (ji,jj,jk-1), bsp (ji,jj,jk-1), &
1059	& csp (ji,jj,jk-1), dsp (ji,jj,jk-1) )
1060	END DO
1061	END DO
1062	END DO
1063
1064	! Z coordinate of U(ji,jj,1:jpkm1) and V(ji,jj,1:jpkm1)
1065
1066	! Prepare zsshu_n and zsshv_n
1067	DO jj = 2, jpjm1
1068	DO ji = 2, jpim1
1069	!!gm BUG ? if it is ssh at u- & v-point then it should be:
1070	! zsshu_n(ji,jj) = (e1e2t(ji,jj) * ssh(ji,jj,Kmm) + e1e2t(ji+1,jj) * ssh(ji+1,jj,Kmm)) * &
1071	! & r1_e1e2u(ji,jj) * umask(ji,jj,1) * 0.5_wp
1072	! zsshv_n(ji,jj) = (e1e2t(ji,jj) * ssh(ji,jj,Kmm) + e1e2t(ji,jj+1) * ssh(ji,jj+1,Kmm)) * &
1073	! & r1_e1e2v(ji,jj) * vmask(ji,jj,1) * 0.5_wp
1074	!!gm not this:
1075	zsshu_n(ji,jj) = (e1e2u(ji,jj) * ssh(ji,jj,Kmm) + e1e2u(ji+1, jj) * ssh(ji+1,jj,Kmm)) * &
1076	& r1_e1e2u(ji,jj) * umask(ji,jj,1) * 0.5_wp
1077	zsshv_n(ji,jj) = (e1e2v(ji,jj) * ssh(ji,jj,Kmm) + e1e2v(ji+1, jj) * ssh(ji,jj+1,Kmm)) * &
1078	& r1_e1e2v(ji,jj) * vmask(ji,jj,1) * 0.5_wp
1079	END DO
1080	END DO
1081
1082	CALL lbc_lnk_multi ('dynhpg', zsshu_n, 'U', 1., zsshv_n, 'V', 1. )
1083
1084	DO jj = 2, jpjm1
1085	DO ji = 2, jpim1
1086	zu(ji,jj,1) = - ( e3u(ji,jj,1,Kmm) - zsshu_n(ji,jj) * znad)
1087	zv(ji,jj,1) = - ( e3v(ji,jj,1,Kmm) - zsshv_n(ji,jj) * znad)
1088	END DO
1089	END DO
1090
1091	DO jk = 2, jpkm1
1092	DO jj = 2, jpjm1
1093	DO ji = 2, jpim1
1094	zu(ji,jj,jk) = zu(ji,jj,jk-1) - e3u(ji,jj,jk,Kmm)
1095	zv(ji,jj,jk) = zv(ji,jj,jk-1) - e3v(ji,jj,jk,Kmm)
1096	END DO
1097	END DO
1098	END DO
1099
1100	DO jk = 1, jpkm1
1101	DO jj = 2, jpjm1
1102	DO ji = 2, jpim1
1103	zu(ji,jj,jk) = zu(ji,jj,jk) + 0.5_wp * e3u(ji,jj,jk,Kmm)
1104	zv(ji,jj,jk) = zv(ji,jj,jk) + 0.5_wp * e3v(ji,jj,jk,Kmm)
1105	END DO
1106	END DO
1107	END DO
1108
1109	DO jk = 1, jpkm1
1110	DO jj = 2, jpjm1
1111	DO ji = 2, jpim1
1112	zu(ji,jj,jk) = MIN( zu(ji,jj,jk) , MAX( -zdept(ji,jj,jk) , -zdept(ji+1,jj,jk) ) )
1113	zu(ji,jj,jk) = MAX( zu(ji,jj,jk) , MIN( -zdept(ji,jj,jk) , -zdept(ji+1,jj,jk) ) )
1114	zv(ji,jj,jk) = MIN( zv(ji,jj,jk) , MAX( -zdept(ji,jj,jk) , -zdept(ji,jj+1,jk) ) )
1115	zv(ji,jj,jk) = MAX( zv(ji,jj,jk) , MIN( -zdept(ji,jj,jk) , -zdept(ji,jj+1,jk) ) )
1116	END DO
1117	END DO
1118	END DO
1119
1120
1121	DO jk = 1, jpkm1
1122	DO jj = 2, jpjm1
1123	DO ji = 2, jpim1
1124	zpwes = 0._wp; zpwed = 0._wp
1125	zpnss = 0._wp; zpnsd = 0._wp
1126	zuijk = zu(ji,jj,jk)
1127	zvijk = zv(ji,jj,jk)
1128
1129	!!!!! for u equation
1130	IF( jk <= mbku(ji,jj) ) THEN
1131	IF( -zdept(ji+1,jj,jk) >= -zdept(ji,jj,jk) ) THEN
1132	jis = ji + 1; jid = ji
1133	ELSE
1134	jis = ji; jid = ji +1
1135	ENDIF
1136
1137	! integrate the pressure on the shallow side
1138	jk1 = jk
1139	DO WHILE ( -zdept(jis,jj,jk1) > zuijk )
1140	IF( jk1 == mbku(ji,jj) ) THEN
1141	zuijk = -zdept(jis,jj,jk1)
1142	EXIT
1143	ENDIF
1144	zdeps = MIN(zdept(jis,jj,jk1+1), -zuijk)
1145	zpwes = zpwes + &
1146	integ_spline(zdept(jis,jj,jk1), zdeps, &
1147	asp(jis,jj,jk1), bsp(jis,jj,jk1), &
1148	csp(jis,jj,jk1), dsp(jis,jj,jk1))
1149	jk1 = jk1 + 1
1150	END DO
1151
1152	! integrate the pressure on the deep side
1153	jk1 = jk
1154	DO WHILE ( -zdept(jid,jj,jk1) < zuijk )
1155	IF( jk1 == 1 ) THEN
1156	zdeps = zdept(jid,jj,1) + MIN(zuijk, ssh(jid,jj,Kmm)*znad)
1157	zrhdt1 = zrhh(jid,jj,1) - interp3(zdept(jid,jj,1), asp(jid,jj,1), &
1158	bsp(jid,jj,1), csp(jid,jj,1), &
1159	dsp(jid,jj,1)) * zdeps
1160	zpwed = zpwed + 0.5_wp * (zrhh(jid,jj,1) + zrhdt1) * zdeps
1161	EXIT
1162	ENDIF
1163	zdeps = MAX(zdept(jid,jj,jk1-1), -zuijk)
1164	zpwed = zpwed + &
1165	integ_spline(zdeps, zdept(jid,jj,jk1), &
1166	asp(jid,jj,jk1-1), bsp(jid,jj,jk1-1), &
1167	csp(jid,jj,jk1-1), dsp(jid,jj,jk1-1) )
1168	jk1 = jk1 - 1
1169	END DO
1170
1171	! update the momentum trends in u direction
1172
1173	zdpdx1 = zcoef0 * r1_e1u(ji,jj) * ( zhpi(ji+1,jj,jk) - zhpi(ji,jj,jk) )
1174	IF( .NOT.ln_linssh ) THEN
1175	zdpdx2 = zcoef0 * r1_e1u(ji,jj) * &
1176	& ( REAL(jis-jid, wp) * (zpwes + zpwed) + (ssh(ji+1,jj,Kmm)-ssh(ji,jj,Kmm)) )
1177	ELSE
1178	zdpdx2 = zcoef0 * r1_e1u(ji,jj) * REAL(jis-jid, wp) * (zpwes + zpwed)
1179	ENDIF
1180	IF( ln_wd_il ) THEN
1181	zdpdx1 = zdpdx1 * zcpx(ji,jj) * wdrampu(ji,jj)
1182	zdpdx2 = zdpdx2 * zcpx(ji,jj) * wdrampu(ji,jj)
1183	ENDIF
1184	puu(ji,jj,jk,Krhs) = puu(ji,jj,jk,Krhs) + (zdpdx1 + zdpdx2) * umask(ji,jj,jk)
1185	ENDIF
1186
1187	!!!!! for v equation
1188	IF( jk <= mbkv(ji,jj) ) THEN
1189	IF( -zdept(ji,jj+1,jk) >= -zdept(ji,jj,jk) ) THEN
1190	jjs = jj + 1; jjd = jj
1191	ELSE
1192	jjs = jj ; jjd = jj + 1
1193	ENDIF
1194
1195	! integrate the pressure on the shallow side
1196	jk1 = jk
1197	DO WHILE ( -zdept(ji,jjs,jk1) > zvijk )
1198	IF( jk1 == mbkv(ji,jj) ) THEN
1199	zvijk = -zdept(ji,jjs,jk1)
1200	EXIT
1201	ENDIF
1202	zdeps = MIN(zdept(ji,jjs,jk1+1), -zvijk)
1203	zpnss = zpnss + &
1204	integ_spline(zdept(ji,jjs,jk1), zdeps, &
1205	asp(ji,jjs,jk1), bsp(ji,jjs,jk1), &
1206	csp(ji,jjs,jk1), dsp(ji,jjs,jk1) )
1207	jk1 = jk1 + 1
1208	END DO
1209
1210	! integrate the pressure on the deep side
1211	jk1 = jk
1212	DO WHILE ( -zdept(ji,jjd,jk1) < zvijk )
1213	IF( jk1 == 1 ) THEN
1214	zdeps = zdept(ji,jjd,1) + MIN(zvijk, ssh(ji,jjd,Kmm)*znad)
1215	zrhdt1 = zrhh(ji,jjd,1) - interp3(zdept(ji,jjd,1), asp(ji,jjd,1), &
1216	bsp(ji,jjd,1), csp(ji,jjd,1), &
1217	dsp(ji,jjd,1) ) * zdeps
1218	zpnsd = zpnsd + 0.5_wp * (zrhh(ji,jjd,1) + zrhdt1) * zdeps
1219	EXIT
1220	ENDIF
1221	zdeps = MAX(zdept(ji,jjd,jk1-1), -zvijk)
1222	zpnsd = zpnsd + &
1223	integ_spline(zdeps, zdept(ji,jjd,jk1), &
1224	asp(ji,jjd,jk1-1), bsp(ji,jjd,jk1-1), &
1225	csp(ji,jjd,jk1-1), dsp(ji,jjd,jk1-1) )
1226	jk1 = jk1 - 1
1227	END DO
1228
1229
1230	! update the momentum trends in v direction
1231
1232	zdpdy1 = zcoef0 * r1_e2v(ji,jj) * ( zhpi(ji,jj+1,jk) - zhpi(ji,jj,jk) )
1233	IF( .NOT.ln_linssh ) THEN
1234	zdpdy2 = zcoef0 * r1_e2v(ji,jj) * &
1235	( REAL(jjs-jjd, wp) * (zpnss + zpnsd) + (ssh(ji,jj+1,Kmm)-ssh(ji,jj,Kmm)) )
1236	ELSE
1237	zdpdy2 = zcoef0 * r1_e2v(ji,jj) * REAL(jjs-jjd, wp) * (zpnss + zpnsd )
1238	ENDIF
1239	IF( ln_wd_il ) THEN
1240	zdpdy1 = zdpdy1 * zcpy(ji,jj) * wdrampv(ji,jj)
1241	zdpdy2 = zdpdy2 * zcpy(ji,jj) * wdrampv(ji,jj)
1242	ENDIF
1243
1244	pvv(ji,jj,jk,Krhs) = pvv(ji,jj,jk,Krhs) + (zdpdy1 + zdpdy2) * vmask(ji,jj,jk)
1245	ENDIF
1246	!
1247	END DO
1248	END DO
1249	END DO
1250	!
1251	IF( ln_wd_il ) DEALLOCATE( zcpx, zcpy )
1252	!
1253	END SUBROUTINE hpg_prj
1254
1255
1256	SUBROUTINE cspline( fsp, xsp, asp, bsp, csp, dsp, polynomial_type )
1257	!!----------------------------------------------------------------------
1258	!! * ROUTINE cspline *
1259	!!
1260	!! ** Purpose : constrained cubic spline interpolation
1261	!!
1262	!! ** Method : f(x) = asp + bspx + cspx^2 + dsp*x^3
1263	!!
1264	!! Reference: CJC Kruger, Constrained Cubic Spline Interpoltation
1265	!!----------------------------------------------------------------------
1266	REAL(wp), DIMENSION(:,:,:), INTENT(in ) :: fsp, xsp ! value and coordinate
1267	REAL(wp), DIMENSION(:,:,:), INTENT( out) :: asp, bsp, csp, dsp ! coefficients of the interpoated function
1268	INTEGER , INTENT(in ) :: polynomial_type ! 1: cubic spline ; 2: Linear
1269	!
1270	INTEGER :: ji, jj, jk ! dummy loop indices
1271	INTEGER :: jpi, jpj, jpkm1
1272	REAL(wp) :: zdf1, zdf2, zddf1, zddf2, ztmp1, ztmp2, zdxtmp
1273	REAL(wp) :: zdxtmp1, zdxtmp2, zalpha
1274	REAL(wp) :: zdf(size(fsp,3))
1275	!!----------------------------------------------------------------------
1276	!
1277	!!gm WHAT !!!!! THIS IS VERY DANGEROUS !!!!!
1278	jpi = size(fsp,1)
1279	jpj = size(fsp,2)
1280	jpkm1 = MAX( 1, size(fsp,3) - 1 )
1281	!
1282	IF (polynomial_type == 1) THEN ! Constrained Cubic Spline
1283	DO ji = 1, jpi
1284	DO jj = 1, jpj
1285	!!Fritsch&Butland's method, 1984 (preferred, but more computation)
1286	! DO jk = 2, jpkm1-1
1287	! zdxtmp1 = xsp(ji,jj,jk) - xsp(ji,jj,jk-1)
1288	! zdxtmp2 = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1289	! zdf1 = ( fsp(ji,jj,jk) - fsp(ji,jj,jk-1) ) / zdxtmp1
1290	! zdf2 = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp2
1291	!
1292	! zalpha = ( zdxtmp1 + 2._wp * zdxtmp2 ) / ( zdxtmp1 + zdxtmp2 ) / 3._wp
1293	!
1294	! IF(zdf1 * zdf2 <= 0._wp) THEN
1295	! zdf(jk) = 0._wp
1296	! ELSE
1297	! zdf(jk) = zdf1 * zdf2 / ( ( 1._wp - zalpha ) * zdf1 + zalpha * zdf2 )
1298	! ENDIF
1299	! END DO
1300
1301	!!Simply geometric average
1302	DO jk = 2, jpkm1-1
1303	zdf1 = (fsp(ji,jj,jk ) - fsp(ji,jj,jk-1)) / (xsp(ji,jj,jk ) - xsp(ji,jj,jk-1))
1304	zdf2 = (fsp(ji,jj,jk+1) - fsp(ji,jj,jk )) / (xsp(ji,jj,jk+1) - xsp(ji,jj,jk ))
1305
1306	IF(zdf1 * zdf2 <= 0._wp) THEN
1307	zdf(jk) = 0._wp
1308	ELSE
1309	zdf(jk) = 2._wp * zdf1 * zdf2 / (zdf1 + zdf2)
1310	ENDIF
1311	END DO
1312
1313	zdf(1) = 1.5_wp * ( fsp(ji,jj,2) - fsp(ji,jj,1) ) / &
1314	& ( xsp(ji,jj,2) - xsp(ji,jj,1) ) - 0.5_wp * zdf(2)
1315	zdf(jpkm1) = 1.5_wp * ( fsp(ji,jj,jpkm1) - fsp(ji,jj,jpkm1-1) ) / &
1316	& ( xsp(ji,jj,jpkm1) - xsp(ji,jj,jpkm1-1) ) - 0.5_wp * zdf(jpkm1 - 1)
1317
1318	DO jk = 1, jpkm1 - 1
1319	zdxtmp = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1320	ztmp1 = (zdf(jk+1) + 2._wp * zdf(jk)) / zdxtmp
1321	ztmp2 = 6._wp * (fsp(ji,jj,jk+1) - fsp(ji,jj,jk)) / zdxtmp / zdxtmp
1322	zddf1 = -2._wp * ztmp1 + ztmp2
1323	ztmp1 = (2._wp * zdf(jk+1) + zdf(jk)) / zdxtmp
1324	zddf2 = 2._wp * ztmp1 - ztmp2
1325
1326	dsp(ji,jj,jk) = (zddf2 - zddf1) / 6._wp / zdxtmp
1327	csp(ji,jj,jk) = ( xsp(ji,jj,jk+1) * zddf1 - xsp(ji,jj,jk)*zddf2 ) / 2._wp / zdxtmp
1328	bsp(ji,jj,jk) = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp - &
1329	& csp(ji,jj,jk) * ( xsp(ji,jj,jk+1) + xsp(ji,jj,jk) ) - &
1330	& dsp(ji,jj,jk) * ((xsp(ji,jj,jk+1) + xsp(ji,jj,jk))**2 - &
1331	& xsp(ji,jj,jk+1) * xsp(ji,jj,jk))
1332	asp(ji,jj,jk) = fsp(ji,jj,jk) - xsp(ji,jj,jk) * (bsp(ji,jj,jk) + &
1333	& (xsp(ji,jj,jk) * (csp(ji,jj,jk) + &
1334	& dsp(ji,jj,jk) * xsp(ji,jj,jk))))
1335	END DO
1336	END DO
1337	END DO
1338
1339	ELSEIF ( polynomial_type == 2 ) THEN ! Linear
1340	DO ji = 1, jpi
1341	DO jj = 1, jpj
1342	DO jk = 1, jpkm1-1
1343	zdxtmp =xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1344	ztmp1 = fsp(ji,jj,jk+1) - fsp(ji,jj,jk)
1345
1346	dsp(ji,jj,jk) = 0._wp
1347	csp(ji,jj,jk) = 0._wp
1348	bsp(ji,jj,jk) = ztmp1 / zdxtmp
1349	asp(ji,jj,jk) = fsp(ji,jj,jk) - bsp(ji,jj,jk) * xsp(ji,jj,jk)
1350	END DO
1351	END DO
1352	END DO
1353	!
1354	ELSE
1355	CALL ctl_stop( 'invalid polynomial type in cspline' )
1356	ENDIF
1357	!
1358	END SUBROUTINE cspline
1359
1360
1361	FUNCTION interp1(x, xl, xr, fl, fr) RESULT(f)
1362	!!----------------------------------------------------------------------
1363	!! * ROUTINE interp1 *
1364	!!
1365	!! ** Purpose : 1-d linear interpolation
1366	!!
1367	!! ** Method : interpolation is straight forward
1368	!! extrapolation is also permitted (no value limit)
1369	!!----------------------------------------------------------------------
1370	REAL(wp), INTENT(in) :: x, xl, xr, fl, fr
1371	REAL(wp) :: f ! result of the interpolation (extrapolation)
1372	REAL(wp) :: zdeltx
1373	!!----------------------------------------------------------------------
1374	!
1375	zdeltx = xr - xl
1376	IF( abs(zdeltx) <= 10._wp * EPSILON(x) ) THEN
1377	f = 0.5_wp * (fl + fr)
1378	ELSE
1379	f = ( (x - xl ) * fr - ( x - xr ) * fl ) / zdeltx
1380	ENDIF
1381	!
1382	END FUNCTION interp1
1383
1384
1385	FUNCTION interp2( x, a, b, c, d ) RESULT(f)
1386	!!----------------------------------------------------------------------
1387	!! * ROUTINE interp1 *
1388	!!
1389	!! ** Purpose : 1-d constrained cubic spline interpolation
1390	!!
1391	!! ** Method : cubic spline interpolation
1392	!!
1393	!!----------------------------------------------------------------------
1394	REAL(wp), INTENT(in) :: x, a, b, c, d
1395	REAL(wp) :: f ! value from the interpolation
1396	!!----------------------------------------------------------------------
1397	!
1398	f = a + x* ( b + x * ( c + d * x ) )
1399	!
1400	END FUNCTION interp2
1401
1402
1403	FUNCTION interp3( x, a, b, c, d ) RESULT(f)
1404	!!----------------------------------------------------------------------
1405	!! * ROUTINE interp1 *
1406	!!
1407	!! ** Purpose : Calculate the first order of derivative of
1408	!! a cubic spline function y=a+bx+cx^2+d*x^3
1409	!!
1410	!! ** Method : f=dy/dx=b+2cx+3dx^2
1411	!!
1412	!!----------------------------------------------------------------------
1413	REAL(wp), INTENT(in) :: x, a, b, c, d
1414	REAL(wp) :: f ! value from the interpolation
1415	!!----------------------------------------------------------------------
1416	!
1417	f = b + x * ( 2._wp * c + 3._wp * d * x)
1418	!
1419	END FUNCTION interp3
1420
1421
1422	FUNCTION integ_spline( xl, xr, a, b, c, d ) RESULT(f)
1423	!!----------------------------------------------------------------------
1424	!! * ROUTINE interp1 *
1425	!!
1426	!! ** Purpose : 1-d constrained cubic spline integration
1427	!!
1428	!! ** Method : integrate polynomial a+bx+cx^2+dx^3 from xl to xr
1429	!!
1430	!!----------------------------------------------------------------------
1431	REAL(wp), INTENT(in) :: xl, xr, a, b, c, d
1432	REAL(wp) :: za1, za2, za3
1433	REAL(wp) :: f ! integration result
1434	!!----------------------------------------------------------------------
1435	!
1436	za1 = 0.5_wp * b
1437	za2 = c / 3.0_wp
1438	za3 = 0.25_wp * d
1439	!
1440	f = xr * ( a + xr * ( za1 + xr * ( za2 + za3 * xr ) ) ) - &
1441	& xl * ( a + xl * ( za1 + xl * ( za2 + za3 * xl ) ) )
1442	!
1443	END FUNCTION integ_spline
1444
1445	!!======================================================================
1446	END MODULE dynhpg
1447

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