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dynhpg.F90

Last change on this file was 8112, checked in by davestorkey, 7 years ago
UKMO/branch/dev_r5518_GO6_package_FVPS: FVPS implementation
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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 sbc_oce ! surface variable (only for the flag with ice shelf)
34	USE dom_oce ! ocean space and time domain
35	USE phycst ! physical constants
36	USE trd_oce ! trends: ocean variables
37	USE trddyn ! trend manager: dynamics
38	!
39	USE in_out_manager ! I/O manager
40	USE prtctl ! Print control
41	USE lbclnk ! lateral boundary condition
42	USE lib_mpp ! MPP library
43	USE eosbn2 ! compute density
44	USE wrk_nemo ! Memory Allocation
45	USE timing ! Timing
46	USE iom
47
48	IMPLICIT NONE
49	PRIVATE
50
51	PUBLIC dyn_hpg ! routine called by step module
52	PUBLIC dyn_hpg_init ! routine called by opa module
53
54	! !!* Namelist namdyn_hpg : hydrostatic pressure gradient
55	LOGICAL , PUBLIC :: ln_hpg_zco !: z-coordinate - full steps
56	LOGICAL , PUBLIC :: ln_hpg_zps !: z-coordinate - partial steps (interpolation)
57	LOGICAL , PUBLIC :: ln_hpg_sco !: s-coordinate (standard jacobian formulation)
58	LOGICAL , PUBLIC :: ln_hpg_djc !: s-coordinate (Density Jacobian with Cubic polynomial)
59	LOGICAL , PUBLIC :: ln_hpg_prj !: s-coordinate (Pressure Jacobian scheme)
60	LOGICAL , PUBLIC :: ln_hpg_isf !: s-coordinate similar to sco modify for isf
61	LOGICAL , PUBLIC :: ln_dynhpg_imp !: semi-implicite hpg flag
62
63	INTEGER , PUBLIC :: nhpg = 0 ! = 0 to 7, type of pressure gradient scheme used ! (deduced from ln_hpg_... flags) (PUBLIC for TAM)
64
65	!! * Substitutions
66	# include "domzgr_substitute.h90"
67	# include "vectopt_loop_substitute.h90"
68	!!----------------------------------------------------------------------
69	!! NEMO/OPA 3.3 , NEMO Consortium (2010)
70	!! $Id$
71	!! Software governed by the CeCILL licence (NEMOGCM/NEMO_CeCILL.txt)
72	!!----------------------------------------------------------------------
73	CONTAINS
74
75	SUBROUTINE dyn_hpg( kt )
76	!!---------------------------------------------------------------------
77	!! * ROUTINE dyn_hpg *
78	!!
79	!! ** Method : Call the hydrostatic pressure gradient routine
80	!! using the scheme defined in the namelist
81	!!
82	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
83	!! - send trends to trd_dyn for futher diagnostics (l_trddyn=T)
84	!!----------------------------------------------------------------------
85	INTEGER, INTENT(in) :: kt ! ocean time-step index
86	REAL(wp), POINTER, DIMENSION(:,:,:) :: ztrdu, ztrdv
87	!!----------------------------------------------------------------------
88	!
89	IF( nn_timing == 1 ) CALL timing_start('dyn_hpg')
90	!
91	IF( l_trddyn ) THEN ! Temporary saving of ua and va trends (l_trddyn)
92	CALL wrk_alloc( jpi,jpj,jpk, ztrdu, ztrdv )
93	ztrdu(:,:,:) = ua(:,:,:)
94	ztrdv(:,:,:) = va(:,:,:)
95	ENDIF
96	!
97	SELECT CASE ( nhpg ) ! Hydrostatic pressure gradient computation
98	CASE ( 0 ) ; CALL hpg_zco ( kt ) ! z-coordinate
99	CASE ( 1 ) ; CALL hpg_zps ( kt ) ! z-coordinate plus partial steps (interpolation)
100	CASE ( 2 ) ; CALL hpg_sco ( kt ) ! s-coordinate (standard jacobian formulation)
101	CASE ( 3 ) ; CALL hpg_djc ( kt ) ! s-coordinate (Density Jacobian with Cubic polynomial)
102	CASE ( 4 ) ; CALL hpg_prj ( kt ) ! s-coordinate (Pressure Jacobian scheme)
103	CASE ( 5 ) ; CALL hpg_isf ( kt ) ! s-coordinate similar to sco modify for ice shelf
104	END SELECT
105	!
106	IF( l_trddyn ) THEN ! save the hydrostatic pressure gradient trends for momentum trend diagnostics
107	ztrdu(:,:,:) = ua(:,:,:) - ztrdu(:,:,:)
108	ztrdv(:,:,:) = va(:,:,:) - ztrdv(:,:,:)
109	CALL trd_dyn( ztrdu, ztrdv, jpdyn_hpg, kt )
110	CALL wrk_dealloc( jpi,jpj,jpk, ztrdu, ztrdv )
111	ENDIF
112	!
113	IF(ln_ctl) CALL prt_ctl( tab3d_1=ua, clinfo1=' hpg - Ua: ', mask1=umask, &
114	& tab3d_2=va, clinfo2= ' Va: ', mask2=vmask, clinfo3='dyn' )
115	!
116	IF( nn_timing == 1 ) CALL timing_stop('dyn_hpg')
117	!
118	END SUBROUTINE dyn_hpg
119
120
121	SUBROUTINE dyn_hpg_init
122	!!----------------------------------------------------------------------
123	!! * ROUTINE dyn_hpg_init *
124	!!
125	!! ** Purpose : initializations for the hydrostatic pressure gradient
126	!! computation and consistency control
127	!!
128	!! ** Action : Read the namelist namdyn_hpg and check the consistency
129	!! with the type of vertical coordinate used (zco, zps, sco)
130	!!----------------------------------------------------------------------
131	INTEGER :: ioptio = 0 ! temporary integer
132	INTEGER :: ios ! Local integer output status for namelist read
133	!!
134	NAMELIST/namdyn_hpg/ ln_hpg_zco, ln_hpg_zps, ln_hpg_sco, &
135	& ln_hpg_djc, ln_hpg_prj, ln_hpg_isf, ln_dynhpg_imp
136	!!----------------------------------------------------------------------
137	!
138	REWIND( numnam_ref ) ! Namelist namdyn_hpg in reference namelist : Hydrostatic pressure gradient
139	READ ( numnam_ref, namdyn_hpg, IOSTAT = ios, ERR = 901)
140	901 IF( ios /= 0 ) CALL ctl_nam ( ios , 'namdyn_hpg in reference namelist', lwp )
141
142	REWIND( numnam_cfg ) ! Namelist namdyn_hpg in configuration namelist : Hydrostatic pressure gradient
143	READ ( numnam_cfg, namdyn_hpg, IOSTAT = ios, ERR = 902 )
144	902 IF( ios /= 0 ) CALL ctl_nam ( ios , 'namdyn_hpg in configuration namelist', lwp )
145	IF(lwm) WRITE ( numond, namdyn_hpg )
146	!
147	IF(lwp) THEN ! Control print
148	WRITE(numout,*)
149	WRITE(numout,*) 'dyn_hpg_init : hydrostatic pressure gradient initialisation'
150	WRITE(numout,*) '~~~~~~~~~~~~'
151	WRITE(numout,*) ' Namelist namdyn_hpg : choice of hpg scheme'
152	WRITE(numout,*) ' z-coord. - full steps ln_hpg_zco = ', ln_hpg_zco
153	WRITE(numout,*) ' z-coord. - partial steps (interpolation) ln_hpg_zps = ', ln_hpg_zps
154	WRITE(numout,*) ' s-coord. (standard jacobian formulation) ln_hpg_sco = ', ln_hpg_sco
155	WRITE(numout,*) ' s-coord. (standard jacobian formulation) for isf ln_hpg_isf = ', ln_hpg_isf
156	WRITE(numout,*) ' s-coord. (Density Jacobian: Cubic polynomial) ln_hpg_djc = ', ln_hpg_djc
157	WRITE(numout,*) ' s-coord. (Pressure Jacobian: Cubic polynomial) ln_hpg_prj = ', ln_hpg_prj
158	WRITE(numout,*) ' time stepping: centered (F) or semi-implicit (T) ln_dynhpg_imp = ', ln_dynhpg_imp
159	ENDIF
160	!
161	IF( ln_hpg_djc ) &
162	& CALL ctl_stop('dyn_hpg_init : Density Jacobian: Cubic polynominal method &
163	& currently disabled (bugs under investigation). Please select &
164	& either ln_hpg_sco or ln_hpg_prj instead')
165	!
166	IF( lk_vvl .AND. .NOT. (ln_hpg_sco.OR.ln_hpg_prj.OR.ln_hpg_isf) ) &
167	& CALL ctl_stop('dyn_hpg_init : variable volume key_vvl requires:&
168	& the standard jacobian formulation hpg_sco or &
169	& the pressure jacobian formulation hpg_prj')
170
171	IF( ln_hpg_isf .AND. .NOT. ln_isfcav ) &
172	& CALL ctl_stop( ' hpg_isf not available if ln_isfcav = false ' )
173	IF( .NOT. ln_hpg_isf .AND. ln_isfcav ) &
174	& CALL ctl_stop( 'Only hpg_isf has been corrected to work with ice shelf cavity.' )
175	!
176	! ! Set nhpg from ln_hpg_... flags
177	IF( ln_hpg_zco ) nhpg = 0
178	IF( ln_hpg_zps ) nhpg = 1
179	IF( ln_hpg_sco ) nhpg = 2
180	IF( ln_hpg_djc ) nhpg = 3
181	IF( ln_hpg_prj ) nhpg = 4
182	IF( ln_hpg_isf ) nhpg = 5
183	!
184	! ! Consistency check
185	ioptio = 0
186	IF( ln_hpg_zco ) ioptio = ioptio + 1
187	IF( ln_hpg_zps ) ioptio = ioptio + 1
188	IF( ln_hpg_sco ) ioptio = ioptio + 1
189	IF( ln_hpg_djc ) ioptio = ioptio + 1
190	IF( ln_hpg_prj ) ioptio = ioptio + 1
191	IF( ln_hpg_isf ) ioptio = ioptio + 1
192	IF( ioptio /= 1 ) CALL ctl_stop( 'NO or several hydrostatic pressure gradient options used' )
193	!
194	! initialisation of ice load
195	riceload(:,:)=0.0
196	!
197	END SUBROUTINE dyn_hpg_init
198
199
200	SUBROUTINE hpg_zco( kt )
201	!!---------------------------------------------------------------------
202	!! * ROUTINE hpg_zco *
203	!!
204	!! ** Method : z-coordinate case, levels are horizontal surfaces.
205	!! The now hydrostatic pressure gradient at a given level, jk,
206	!! is computed by taking the vertical integral of the in-situ
207	!! density gradient along the model level from the suface to that
208	!! level: zhpi = grav .....
209	!! zhpj = grav .....
210	!! add it to the general momentum trend (ua,va).
211	!! ua = ua - 1/e1u * zhpi
212	!! va = va - 1/e2v * zhpj
213	!!
214	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
215	!!----------------------------------------------------------------------
216	INTEGER, INTENT(in) :: kt ! ocean time-step index
217	!!
218	INTEGER :: ji, jj, jk ! dummy loop indices
219	REAL(wp) :: zcoef0, zcoef1 ! temporary scalars
220	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj
221	!!----------------------------------------------------------------------
222	!
223	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zhpj )
224	!
225	IF( kt == nit000 ) THEN
226	IF(lwp) WRITE(numout,*)
227	IF(lwp) WRITE(numout,*) 'dyn:hpg_zco : hydrostatic pressure gradient trend'
228	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ z-coordinate case '
229	ENDIF
230
231	zcoef0 = - grav * 0.5_wp ! Local constant initialization
232
233	! Surface value
234	DO jj = 2, jpjm1
235	DO ji = fs_2, fs_jpim1 ! vector opt.
236	zcoef1 = zcoef0 * fse3w(ji,jj,1)
237	! hydrostatic pressure gradient
238	zhpi(ji,jj,1) = zcoef1 * ( rhd(ji+1,jj,1) - rhd(ji,jj,1) ) / e1u(ji,jj)
239	zhpj(ji,jj,1) = zcoef1 * ( rhd(ji,jj+1,1) - rhd(ji,jj,1) ) / e2v(ji,jj)
240	! add to the general momentum trend
241	ua(ji,jj,1) = ua(ji,jj,1) + zhpi(ji,jj,1)
242	va(ji,jj,1) = va(ji,jj,1) + zhpj(ji,jj,1)
243	END DO
244	END DO
245
246	!
247	! interior value (2=<jk=<jpkm1)
248	DO jk = 2, jpkm1
249	DO jj = 2, jpjm1
250	DO ji = fs_2, fs_jpim1 ! vector opt.
251	zcoef1 = zcoef0 * fse3w(ji,jj,jk)
252	! hydrostatic pressure gradient
253	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
254	& + zcoef1 * ( ( rhd(ji+1,jj,jk)+rhd(ji+1,jj,jk-1) ) &
255	& - ( rhd(ji ,jj,jk)+rhd(ji ,jj,jk-1) ) ) / e1u(ji,jj)
256
257	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
258	& + zcoef1 * ( ( rhd(ji,jj+1,jk)+rhd(ji,jj+1,jk-1) ) &
259	& - ( rhd(ji,jj, jk)+rhd(ji,jj ,jk-1) ) ) / e2v(ji,jj)
260	! add to the general momentum trend
261	ua(ji,jj,jk) = ua(ji,jj,jk) + zhpi(ji,jj,jk)
262	va(ji,jj,jk) = va(ji,jj,jk) + zhpj(ji,jj,jk)
263	END DO
264	END DO
265	END DO
266	!
267	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zhpj )
268	!
269	END SUBROUTINE hpg_zco
270
271
272	SUBROUTINE hpg_zps( kt )
273	!!---------------------------------------------------------------------
274	!! * ROUTINE hpg_zps *
275	!!
276	!! ** Method : z-coordinate plus partial steps case. blahblah...
277	!!
278	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
279	!!----------------------------------------------------------------------
280	INTEGER, INTENT(in) :: kt ! ocean time-step index
281	!!
282	INTEGER :: ji, jj, jk ! dummy loop indices
283	INTEGER :: iku, ikv ! temporary integers
284	REAL(wp) :: zcoef0, zcoef1, zcoef2, zcoef3 ! temporary scalars
285	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj
286	!!----------------------------------------------------------------------
287	!
288	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zhpj )
289	!
290	IF( kt == nit000 ) THEN
291	IF(lwp) WRITE(numout,*)
292	IF(lwp) WRITE(numout,*) 'dyn:hpg_zps : hydrostatic pressure gradient trend'
293	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ z-coordinate with partial steps - vector optimization'
294	ENDIF
295
296
297	! Local constant initialization
298	zcoef0 = - grav * 0.5_wp
299
300	! Surface value (also valid in partial step case)
301	DO jj = 2, jpjm1
302	DO ji = fs_2, fs_jpim1 ! vector opt.
303	zcoef1 = zcoef0 * fse3w(ji,jj,1)
304	! hydrostatic pressure gradient
305	zhpi(ji,jj,1) = zcoef1 * ( rhd(ji+1,jj ,1) - rhd(ji,jj,1) ) / e1u(ji,jj)
306	zhpj(ji,jj,1) = zcoef1 * ( rhd(ji ,jj+1,1) - rhd(ji,jj,1) ) / e2v(ji,jj)
307	! add to the general momentum trend
308	ua(ji,jj,1) = ua(ji,jj,1) + zhpi(ji,jj,1)
309	va(ji,jj,1) = va(ji,jj,1) + zhpj(ji,jj,1)
310	END DO
311	END DO
312
313
314	! interior value (2=<jk=<jpkm1)
315	DO jk = 2, jpkm1
316	DO jj = 2, jpjm1
317	DO ji = fs_2, fs_jpim1 ! vector opt.
318	zcoef1 = zcoef0 * fse3w(ji,jj,jk)
319	! hydrostatic pressure gradient
320	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
321	& + zcoef1 * ( ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) ) &
322	& - ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) ) ) / e1u(ji,jj)
323
324	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
325	& + zcoef1 * ( ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) ) &
326	& - ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) ) ) / e2v(ji,jj)
327	! add to the general momentum trend
328	ua(ji,jj,jk) = ua(ji,jj,jk) + zhpi(ji,jj,jk)
329	va(ji,jj,jk) = va(ji,jj,jk) + zhpj(ji,jj,jk)
330	END DO
331	END DO
332	END DO
333
334
335	! partial steps correction at the last level (use gru & grv computed in zpshde.F90)
336	DO jj = 2, jpjm1
337	DO ji = 2, jpim1
338	iku = mbku(ji,jj)
339	ikv = mbkv(ji,jj)
340	zcoef2 = zcoef0 * MIN( fse3w(ji,jj,iku), fse3w(ji+1,jj ,iku) )
341	zcoef3 = zcoef0 * MIN( fse3w(ji,jj,ikv), fse3w(ji ,jj+1,ikv) )
342	IF( iku > 1 ) THEN ! on i-direction (level 2 or more)
343	ua (ji,jj,iku) = ua(ji,jj,iku) - zhpi(ji,jj,iku) ! subtract old value
344	zhpi(ji,jj,iku) = zhpi(ji,jj,iku-1) & ! compute the new one
345	& + zcoef2 * ( rhd(ji+1,jj,iku-1) - rhd(ji,jj,iku-1) + gru(ji,jj) ) / e1u(ji,jj)
346	ua (ji,jj,iku) = ua(ji,jj,iku) + zhpi(ji,jj,iku) ! add the new one to the general momentum trend
347	ENDIF
348	IF( ikv > 1 ) THEN ! on j-direction (level 2 or more)
349	va (ji,jj,ikv) = va(ji,jj,ikv) - zhpj(ji,jj,ikv) ! subtract old value
350	zhpj(ji,jj,ikv) = zhpj(ji,jj,ikv-1) & ! compute the new one
351	& + zcoef3 * ( rhd(ji,jj+1,ikv-1) - rhd(ji,jj,ikv-1) + grv(ji,jj) ) / e2v(ji,jj)
352	va (ji,jj,ikv) = va(ji,jj,ikv) + zhpj(ji,jj,ikv) ! add the new one to the general momentum trend
353	ENDIF
354	END DO
355	END DO
356	!
357	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zhpj )
358	!
359	END SUBROUTINE hpg_zps
360
361	SUBROUTINE hpg_sco( kt )
362	!!---------------------------------------------------------------------
363	!! * ROUTINE hpg_sco *
364	!!
365	!! ** Method : s-coordinate case. Jacobian scheme.
366	!! The now hydrostatic pressure gradient at a given level, jk,
367	!! is computed by taking the vertical integral of the in-situ
368	!! density gradient along the model level from the suface to that
369	!! level. s-coordinates (ln_sco): a corrective term is added
370	!! to the horizontal pressure gradient :
371	!! zhpi = grav ..... + 1/e1u mi(rhd) di[ grav dep3w ]
372	!! zhpj = grav ..... + 1/e2v mj(rhd) dj[ grav dep3w ]
373	!! add it to the general momentum trend (ua,va).
374	!! ua = ua - 1/e1u * zhpi
375	!! va = va - 1/e2v * zhpj
376	!!
377	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
378	!!----------------------------------------------------------------------
379	INTEGER, INTENT(in) :: kt ! ocean time-step index
380	!!
381	INTEGER :: ji, jj, jk ! dummy loop indices
382	REAL(wp) :: zcoef0, zuap, zvap, znad, zvnum ! temporary scalars
383	REAL(wp), POINTER, DIMENSION(:,:) :: zunum, zudenom ! temporary numerators for u and v velocities (for debugging)
384	REAL(wp), POINTER, DIMENSION(:,:) :: zpb, zpt ! pressure at top and bottom of cells
385	REAL(wp), POINTER, DIMENSION(:,:) :: zzb, zzt ! geopotential height (gz) at top and bottom of cells
386	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj, zp3d, zz3d
387	!!----------------------------------------------------------------------
388	!
389	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zhpj, zp3d, zz3d )
390	CALL wrk_alloc( jpi,jpj, zunum, zudenom, zpb, zpt, zzb, zzt )
391	!
392	IF( kt == nit000 ) THEN
393	IF(lwp) WRITE(numout,*)
394	IF(lwp) WRITE(numout,*) 'dyn:hpg_sco : hydrostatic pressure gradient trend'
395	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, OPA original scheme used'
396	ENDIF
397
398	! Local constant initialization
399	zcoef0 = - grav * 0.5_wp
400	! To use density and not density anomaly
401	IF ( lk_vvl ) THEN ; znad = 1._wp ! Variable volume
402	ELSE ; znad = 0._wp ! Fixed volume
403	ENDIF
404
405	! set the pressure values to zero at the top of the model (the atmospheric surface pressure is included in the barotropic step ?)
406	zpb(:,:) = 0._wp
407	zzb(:,:) = grav * sshn(:,:) ! surface height had been incorrectly set to zero; corrected on 24 Apr 2017
408
409	if(lwp) write(numout,*) 'hpg_sco: grav is ',grav
410
411	DO jk = 1, jpkm1
412
413	zpt(:,:) = zpb(:,:)
414	zzt(:,:) = zzb(:,:)
415
416	zp3d(:,:,jk) = zpt(:,:)
417	zz3d(:,:,jk) = zzt(:,:)
418
419	! integrate the pressure from the top of the cell to the bottom of the cell; a factor rho_0 has been removed from the pressure
420	! corrected to calculate for loops from 1 to jpj and 1 to jpi 25 Apr 2017
421	DO jj = 1, jpj
422	DO ji = 1, jpi ! vector opt.
423	!!$ zpb(ji,jj) = zpt(ji,jj) + grav * fse3w_n(ji,jj,jk) * ( znad + rhd(ji,jj,jk) )
424	zpb(ji,jj) = zpt(ji,jj) + grav * fse3t_n(ji,jj,jk) * ( znad + rhd(ji,jj,jk) )
425	zzb(ji,jj) = grav * ( sshn(ji,jj) - fsdepw_n(ji,jj,jk+1) ) ! surface height had been set to zero; corrected on 24 Apr 2017
426	END DO
427	END DO
428
429	! calculate the Jacobian of the pressure forces
430	DO jj = 2, jpjm1
431	DO ji = fs_2, fs_jpim1 ! vector opt.
432	zunum(ji, jj) = ( zpb(ji+1,jj) - zpt(ji , jj) )( zzb(ji , jj) - zzt(ji+1, jj) ) - ( zpb(ji , jj) - zpt(ji+1, jj) )( zzb(ji+1, jj) - zzt(ji , jj) )
433	zudenom(ji, jj) = zpb(ji+1, jj) - zpt(ji , jj) + zpb(ji , jj) - zpt(ji+1, jj)
434	zhpi(ji, jj, jk) = zunum(ji, jj) * r1_e1u(ji, jj) / zudenom(ji, jj)
435	ua(ji,jj,jk) = ua(ji,jj,jk) + zhpi(ji,jj,jk)
436
437	zvnum = ( zpb(ji,jj+1) - zpt(ji, jj ) )( zzb(ji, jj ) - zzt(ji, jj+1) ) - ( zpb(ji, jj ) - zpt(ji, jj+1) )( zzb(ji, jj+1) - zzt(ji, jj ) )
438	zhpj(ji, jj, jk) = zvnum * r1_e2v(ji, jj) / ( zpb(ji, jj+1) - zpt(ji, jj ) + zpb(ji, jj ) - zpt(ji, jj+1) )
439	va(ji,jj,jk) = va(ji,jj,jk) + zhpj(ji,jj,jk)
440	END DO
441	END DO
442
443	! temporary diagnostic outputs (to be used on just a few timesteps on a small configuration)
444	!!$ DO jj = 1, jpj
445	!!$ write(numout, *) 'fse3w_n, jk, jj = ', jk, jj, (fse3w_n(ji,jj,jk), ji = 1, jpi)
446	!!$ write(numout, *) 'rhd, jk, jj = ', jk, jj, (rhd(ji,jj,jk), ji = 1, jpi)
447	!!$ write(numout, *) 'gdepw_n, jk+1, jj = ', jk+1, jj, (gdepw_n(ji,jj,jk+1), ji = 1, jpim1)
448	!!$ write(numout, *) 'r1_e1u, jk, jj = ', jk, jj, (r1_e1u(ji,jj), ji = 1, jpim1)
449	!!$ write(numout, *) 'zpt, jk, jj = ', jk, jj, (zpt(ji,jj), ji = 1, jpi)
450	!!$ write(numout, *) 'zpb, jk, jj = ', jk, jj, (zpb(ji,jj), ji = 1, jpi)
451	!!$ write(numout, *) 'zzt, jk, jj = ', jk, jj, (zzt(ji,jj), ji = 1, jpi)
452	!!$ write(numout, *) 'zzb, jk, jj = ', jk, jj, (zzb(ji,jj), ji = 1, jpi)
453	!!$ write(numout, *) 'zunum, jk, jj = ', jk, jj, (zunum(ji,jj), ji = 1, jpi)
454	!!$ write(numout, *) 'zudenom, jk, jj = ', jk, jj, (zudenom(ji,jj), ji = 1, jpi)
455	!!$ write(numout, *) 'tmask, jk, jj = ', jk, jj, (tmask(ji,jj,jk), ji = 1, jpi)
456	!!$ write(numout, *) 'umask, jk, jj = ', jk, jj, (umask(ji,jj,jk), ji = 1, jpi)
457	!!$ write(numout, *) 'vmask, jk, jj = ', jk, jj, (vmask(ji,jj,jk), ji = 1, jpi)
458	!!$
459	!!$ write(numout, *) 'zhpi, jk, jj = ', jk, jj, (zhpi(ji,jj,jk), ji = 1, jpi)
460	!!$ write(numout, *) 'zhpj, jk, jj = ', jk, jj, (zhpj(ji,jj,jk), ji = 1, jpi)
461	!!$ END DO
462
463	END DO
464	!
465	CALL iom_put("pressure" , zp3d(:,:,:) )
466	CALL iom_put("hpgdepth" , zz3d(:,:,:) )
467	CALL iom_put("rho" , rhd(:,:,:) )
468	CALL iom_put("hpgi", zhpi(:,:,:) )
469	CALL iom_put("hpgj", zhpj(:,:,:) )
470	!
471	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zhpj, zp3d, zz3d )
472	CALL wrk_dealloc( jpi,jpj, zpb, zpt, zzb, zzt )
473	!
474	END SUBROUTINE hpg_sco
475
476	SUBROUTINE hpg_isf( kt )
477	!!---------------------------------------------------------------------
478	!! * ROUTINE hpg_sco *
479	!!
480	!! ** Method : s-coordinate case. Jacobian scheme.
481	!! The now hydrostatic pressure gradient at a given level, jk,
482	!! is computed by taking the vertical integral of the in-situ
483	!! density gradient along the model level from the suface to that
484	!! level. s-coordinates (ln_sco): a corrective term is added
485	!! to the horizontal pressure gradient :
486	!! zhpi = grav ..... + 1/e1u mi(rhd) di[ grav dep3w ]
487	!! zhpj = grav ..... + 1/e2v mj(rhd) dj[ grav dep3w ]
488	!! add it to the general momentum trend (ua,va).
489	!! ua = ua - 1/e1u * zhpi
490	!! va = va - 1/e2v * zhpj
491	!! iceload is added and partial cell case are added to the top and bottom
492	!!
493	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
494	!!----------------------------------------------------------------------
495	INTEGER, INTENT(in) :: kt ! ocean time-step index
496	!!
497	INTEGER :: ji, jj, jk, iku, ikv, ikt, iktp1i, iktp1j ! dummy loop indices
498	REAL(wp) :: zcoef0, zuap, zvap, znad, ze3wu, ze3wv, zuapint, zvapint, zhpjint, zhpiint, zdzwt, zdzwtjp1, zdzwtip1 ! temporary scalars
499	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj, zrhd
500	REAL(wp), POINTER, DIMENSION(:,:,:) :: ztstop
501	REAL(wp), POINTER, DIMENSION(:,:) :: ze3w, zp, zrhdtop_isf, zrhdtop_oce, ziceload, zdept, zpshpi, zpshpj
502	!!----------------------------------------------------------------------
503	!
504	CALL wrk_alloc( jpi,jpj, 2, ztstop)
505	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zhpj, zrhd)
506	CALL wrk_alloc( jpi,jpj, ze3w, zp, zrhdtop_isf, zrhdtop_oce, ziceload, zdept, zpshpi, zpshpj)
507	!
508	IF( kt == nit000 ) THEN
509	IF(lwp) WRITE(numout,*)
510	IF(lwp) WRITE(numout,*) 'dyn:hpg_isf : hydrostatic pressure gradient trend for ice shelf'
511	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, OPA original scheme used'
512	ENDIF
513
514	! Local constant initialization
515	zcoef0 = - grav * 0.5_wp
516	! To use density and not density anomaly
517	! IF ( lk_vvl ) THEN ; znad = 1._wp ! Variable volume
518	! ELSE ; znad = 0._wp ! Fixed volume
519	! ENDIF
520	znad=1._wp
521	! iniitialised to 0. zhpi zhpi
522	zhpi(:,:,:)=0._wp ; zhpj(:,:,:)=0._wp
523
524	!==================================================================================
525	!=====Compute iceload and contribution of the half first wet layer =================
526	!===================================================================================
527
528	! assume water displaced by the ice shelf is at T=-1.9 and S=34.4 (rude)
529	ztstop(:,:,1)=-1.9_wp ; ztstop(:,:,2)=34.4_wp
530
531	! compute density of the water displaced by the ice shelf
532	zrhd = rhd ! save rhd
533	DO jk = 1, jpk
534	zdept(:,:)=gdept_1d(jk)
535	CALL eos(ztstop(:,:,:),zdept(:,:),rhd(:,:,jk))
536	END DO
537	WHERE ( tmask(:,:,:) == 1._wp)
538	rhd(:,:,:) = zrhd(:,:,:) ! replace wet cell by the saved rhd
539	END WHERE
540
541	! compute rhd at the ice/oce interface (ice shelf side)
542	CALL eos(ztstop,risfdep,zrhdtop_isf)
543
544	! compute rhd at the ice/oce interface (ocean side)
545	DO ji=1,jpi
546	DO jj=1,jpj
547	ikt=mikt(ji,jj)
548	ztstop(ji,jj,1)=tsn(ji,jj,ikt,1)
549	ztstop(ji,jj,2)=tsn(ji,jj,ikt,2)
550	END DO
551	END DO
552	CALL eos(ztstop,risfdep,zrhdtop_oce)
553	!
554	! Surface value + ice shelf gradient
555	! compute pressure due to ice shelf load (used to compute hpgi/j for all the level from 1 to miku/v)
556	ziceload = 0._wp
557	DO jj = 1, jpj
558	DO ji = 1, jpi ! vector opt.
559	ikt=mikt(ji,jj)
560	ziceload(ji,jj) = ziceload(ji,jj) + (znad + rhd(ji,jj,1) ) * fse3w(ji,jj,1) * (1._wp - tmask(ji,jj,1))
561	DO jk=2,ikt-1
562	ziceload(ji,jj) = ziceload(ji,jj) + (2._wp * znad + rhd(ji,jj,jk-1) + rhd(ji,jj,jk)) * fse3w(ji,jj,jk) &
563	& * (1._wp - tmask(ji,jj,jk))
564	END DO
565	IF (ikt .GE. 2) ziceload(ji,jj) = ziceload(ji,jj) + (2._wp * znad + zrhdtop_isf(ji,jj) + rhd(ji,jj,ikt-1)) &
566	& * ( risfdep(ji,jj) - gdept_1d(ikt-1) )
567	END DO
568	END DO
569	riceload(:,:) = 0.0_wp ; riceload(:,:)=ziceload(:,:) ! need to be saved for diaar5
570	! compute zp from z=0 to first T wet point (correction due to zps not yet applied)
571	DO jj = 2, jpjm1
572	DO ji = fs_2, fs_jpim1 ! vector opt.
573	ikt=mikt(ji,jj) ; iktp1i=mikt(ji+1,jj); iktp1j=mikt(ji,jj+1)
574	! hydrostatic pressure gradient along s-surfaces and ice shelf pressure
575	! we assume ISF is in isostatic equilibrium
576	zhpi(ji,jj,1) = zcoef0 / e1u(ji,jj) * ( 0.5_wp * fse3w(ji+1,jj ,iktp1i) &
577	& * ( 2._wp * znad + rhd(ji+1,jj ,iktp1i) + zrhdtop_oce(ji+1,jj ) ) &
578	& - 0.5_wp * fse3w(ji ,jj ,ikt ) &
579	& * ( 2._wp * znad + rhd(ji ,jj ,ikt ) + zrhdtop_oce(ji ,jj ) ) &
580	& + ( ziceload(ji+1,jj) - ziceload(ji,jj)) )
581	zhpj(ji,jj,1) = zcoef0 / e2v(ji,jj) * ( 0.5_wp * fse3w(ji ,jj+1,iktp1j) &
582	& * ( 2._wp * znad + rhd(ji ,jj+1,iktp1j) + zrhdtop_oce(ji ,jj+1) ) &
583	& - 0.5_wp * fse3w(ji ,jj ,ikt ) &
584	& * ( 2._wp * znad + rhd(ji ,jj ,ikt ) + zrhdtop_oce(ji ,jj ) ) &
585	& + ( ziceload(ji,jj+1) - ziceload(ji,jj) ) )
586	! s-coordinate pressure gradient correction (=0 if z coordinate)
587	zuap = -zcoef0 * ( rhd (ji+1,jj,1) + rhd (ji,jj,1) + 2._wp * znad ) &
588	& * ( fsde3w(ji+1,jj,1) - fsde3w(ji,jj,1) ) / e1u(ji,jj)
589	zvap = -zcoef0 * ( rhd (ji,jj+1,1) + rhd (ji,jj,1) + 2._wp * znad ) &
590	& * ( fsde3w(ji,jj+1,1) - fsde3w(ji,jj,1) ) / e2v(ji,jj)
591	! add to the general momentum trend
592	ua(ji,jj,1) = ua(ji,jj,1) + (zhpi(ji,jj,1) + zuap) * umask(ji,jj,1)
593	va(ji,jj,1) = va(ji,jj,1) + (zhpj(ji,jj,1) + zvap) * vmask(ji,jj,1)
594	END DO
595	END DO
596	!==================================================================================
597	!===== Compute partial cell contribution for the top cell =========================
598	!==================================================================================
599	DO jj = 2, jpjm1
600	DO ji = fs_2, fs_jpim1 ! vector opt.
601	iku = miku(ji,jj) ;
602	zpshpi(ji,jj)=0.0_wp ; zpshpj(ji,jj)=0.0_wp
603	ze3wu = (gdepw_0(ji+1,jj,iku+1) - gdept_0(ji+1,jj,iku)) - (gdepw_0(ji,jj,iku+1) - gdept_0(ji,jj,iku))
604	! u direction
605	IF ( iku .GT. 1 ) THEN
606	! case iku
607	zhpi(ji,jj,iku) = zcoef0 / e1u(ji,jj) * ze3wu &
608	& * ( rhd (ji+1,jj,iku) + rhd (ji,jj,iku) &
609	& + SIGN(1._wp,ze3wu) * grui(ji,jj) + 2._wp * znad )
610	! corrective term ( = 0 if z coordinate )
611	zuap = -zcoef0 * ( arui(ji,jj) + 2._wp * znad ) * gzui(ji,jj) / e1u(ji,jj)
612	! zhpi will be added in interior loop
613	ua(ji,jj,iku) = ua(ji,jj,iku) + zuap
614	! in case of 2 cell water column, need to save the pressure gradient to compute the bottom pressure
615	IF (mbku(ji,jj) == iku + 1) zpshpi(ji,jj) = zhpi(ji,jj,iku)
616
617	! case iku + 1 (remove the zphi term added in the interior loop and compute the one corrected for zps)
618	zhpiint = zcoef0 / e1u(ji,jj) &
619	& * ( fse3w(ji+1,jj ,iku+1) * ( (rhd(ji+1,jj,iku+1) + znad) &
620	& + (rhd(ji+1,jj,iku ) + znad) ) * tmask(ji+1,jj,iku) &
621	& - fse3w(ji ,jj ,iku+1) * ( (rhd(ji ,jj,iku+1) + znad) &
622	& + (rhd(ji ,jj,iku ) + znad) ) * tmask(ji ,jj,iku) )
623	zhpi(ji,jj,iku+1) = zcoef0 / e1u(ji,jj) * ge3rui(ji,jj) - zhpiint
624	END IF
625
626	! v direction
627	ikv = mikv(ji,jj)
628	ze3wv = (gdepw_0(ji,jj+1,ikv+1) - gdept_0(ji,jj+1,ikv)) - (gdepw_0(ji,jj,ikv+1) - gdept_0(ji,jj,ikv))
629	IF ( ikv .GT. 1 ) THEN
630	! case ikv
631	zhpj(ji,jj,ikv) = zcoef0 / e2v(ji,jj) * ze3wv &
632	& * ( rhd(ji,jj+1,ikv) + rhd (ji,jj,ikv) &
633	& + SIGN(1._wp,ze3wv) * grvi(ji,jj) + 2._wp * znad )
634	! corrective term ( = 0 if z coordinate )
635	zvap = -zcoef0 * ( arvi(ji,jj) + 2._wp * znad ) * gzvi(ji,jj) / e2v(ji,jj)
636	! zhpi will be added in interior loop
637	va(ji,jj,ikv) = va(ji,jj,ikv) + zvap
638	! in case of 2 cell water column, need to save the pressure gradient to compute the bottom pressure
639	IF (mbkv(ji,jj) == ikv + 1) zpshpj(ji,jj) = zhpj(ji,jj,ikv)
640
641	! case ikv + 1 (remove the zphj term added in the interior loop and compute the one corrected for zps)
642	zhpjint = zcoef0 / e2v(ji,jj) &
643	& * ( fse3w(ji ,jj+1,ikv+1) * ( (rhd(ji,jj+1,ikv+1) + znad) &
644	& + (rhd(ji,jj+1,ikv ) + znad) ) * tmask(ji,jj+1,ikv) &
645	& - fse3w(ji ,jj ,ikv+1) * ( (rhd(ji,jj ,ikv+1) + znad) &
646	& + (rhd(ji,jj ,ikv ) + znad) ) * tmask(ji,jj ,ikv) )
647	zhpj(ji,jj,ikv+1) = zcoef0 / e2v(ji,jj) * ge3rvi(ji,jj) - zhpjint
648	END IF
649	END DO
650	END DO
651
652	!==================================================================================
653	!===== Compute interior value =====================================================
654	!==================================================================================
655
656	DO jj = 2, jpjm1
657	DO ji = fs_2, fs_jpim1 ! vector opt.
658	iku=miku(ji,jj); ikv=mikv(ji,jj)
659	DO jk = 2, jpkm1
660	! hydrostatic pressure gradient along s-surfaces
661	! zhpi is masked for the first wet cell (contribution already done in the upper bloc)
662	zhpi(ji,jj,jk) = zhpi(ji,jj,jk) + zhpi(ji,jj,jk-1) &
663	& + zcoef0 / e1u(ji,jj) &
664	& * ( fse3w(ji+1,jj ,jk) * ( (rhd(ji+1,jj,jk ) + znad) &
665	& + (rhd(ji+1,jj,jk-1) + znad) ) * tmask(ji+1,jj,jk-1) &
666	& - fse3w(ji ,jj ,jk) * ( (rhd(ji ,jj,jk ) + znad) &
667	& + (rhd(ji ,jj,jk-1) + znad) ) * tmask(ji ,jj,jk-1) )
668	! s-coordinate pressure gradient correction
669	! corrective term, we mask this term for the first wet level beneath the ice shelf (contribution done in the upper bloc)
670	zuap = - zcoef0 * ( rhd (ji+1,jj ,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
671	& * ( fsde3w(ji+1,jj ,jk) - fsde3w(ji,jj,jk) ) / e1u(ji,jj) * umask(ji,jj,jk-1)
672	ua(ji,jj,jk) = ua(ji,jj,jk) + ( zhpi(ji,jj,jk) + zuap) * umask(ji,jj,jk)
673
674	! hydrostatic pressure gradient along s-surfaces
675	! zhpi is masked for the first wet cell (contribution already done in the upper bloc)
676	zhpj(ji,jj,jk) = zhpj(ji,jj,jk) + zhpj(ji,jj,jk-1) &
677	& + zcoef0 / e2v(ji,jj) &
678	& * ( fse3w(ji ,jj+1,jk) * ( (rhd(ji,jj+1,jk ) + znad) &
679	& + (rhd(ji,jj+1,jk-1) + znad) ) * tmask(ji,jj+1,jk-1) &
680	& - fse3w(ji ,jj ,jk) * ( (rhd(ji,jj ,jk ) + znad) &
681	& + (rhd(ji,jj ,jk-1) + znad) ) * tmask(ji,jj ,jk-1) )
682	! s-coordinate pressure gradient correction
683	! corrective term, we mask this term for the first wet level beneath the ice shelf (contribution done in the upper bloc)
684	zvap = - zcoef0 * ( rhd (ji ,jj+1,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
685	& * ( fsde3w(ji ,jj+1,jk) - fsde3w(ji,jj,jk) ) / e2v(ji,jj) * vmask(ji,jj,jk-1)
686	! add to the general momentum trend
687	va(ji,jj,jk) = va(ji,jj,jk) + ( zhpj(ji,jj,jk) + zvap ) * vmask(ji,jj,jk)
688	END DO
689	END DO
690	END DO
691
692	!==================================================================================
693	!===== Compute bottom cell contribution (partial cell) ============================
694	!==================================================================================
695
696	DO jj = 2, jpjm1
697	DO ji = 2, jpim1
698	iku = mbku(ji,jj)
699	ikv = mbkv(ji,jj)
700
701	IF (iku .GT. 1) THEN
702	! remove old value (interior case)
703	zuap = -zcoef0 * ( rhd (ji+1,jj ,iku) + rhd (ji,jj,iku) + 2._wp * znad ) &
704	& * ( fsde3w(ji+1,jj ,iku) - fsde3w(ji,jj,iku) ) / e1u(ji,jj)
705	ua(ji,jj,iku) = ua(ji,jj,iku) - zhpi(ji,jj,iku) - zuap
706	! put new value
707	! -zpshpi to avoid double contribution of the partial step in the top layer
708	zuap = -zcoef0 * ( aru(ji,jj) + 2._wp * znad ) * gzu(ji,jj) / e1u(ji,jj)
709	zhpi(ji,jj,iku) = zhpi(ji,jj,iku-1) + zcoef0 / e1u(ji,jj) * ge3ru(ji,jj) - zpshpi(ji,jj)
710	ua(ji,jj,iku) = ua(ji,jj,iku) + zhpi(ji,jj,iku) + zuap
711	END IF
712	! v direction
713	IF (ikv .GT. 1) THEN
714	! remove old value (interior case)
715	zvap = -zcoef0 * ( rhd (ji ,jj+1,ikv) + rhd (ji,jj,ikv) + 2._wp * znad ) &
716	& * ( fsde3w(ji ,jj+1,ikv) - fsde3w(ji,jj,ikv) ) / e2v(ji,jj)
717	va(ji,jj,ikv) = va(ji,jj,ikv) - zhpj(ji,jj,ikv) - zvap
718	! put new value
719	! -zpshpj to avoid double contribution of the partial step in the top layer
720	zvap = -zcoef0 * ( arv(ji,jj) + 2._wp * znad ) * gzv(ji,jj) / e2v(ji,jj)
721	zhpj(ji,jj,ikv) = zhpj(ji,jj,ikv-1) + zcoef0 / e2v(ji,jj) * ge3rv(ji,jj) - zpshpj(ji,jj)
722	va(ji,jj,ikv) = va(ji,jj,ikv) + zhpj(ji,jj,ikv) + zvap
723	END IF
724	END DO
725	END DO
726
727	! set back to original density value into the ice shelf cell (maybe useless because it is masked)
728	rhd = zrhd
729	!
730	CALL wrk_dealloc( jpi,jpj,2, ztstop)
731	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zhpj, zrhd)
732	CALL wrk_dealloc( jpi,jpj, ze3w, zp, zrhdtop_isf, zrhdtop_oce, ziceload, zdept, zpshpi, zpshpj)
733	!
734	END SUBROUTINE hpg_isf
735
736
737	SUBROUTINE hpg_djc( kt )
738	!!---------------------------------------------------------------------
739	!! * ROUTINE hpg_djc *
740	!!
741	!! ** Method : Density Jacobian with Cubic polynomial scheme
742	!!
743	!! Reference: Shchepetkin and McWilliams, J. Geophys. Res., 108(C3), 3090, 2003
744	!!----------------------------------------------------------------------
745	INTEGER, INTENT(in) :: kt ! ocean time-step index
746	!!
747	INTEGER :: ji, jj, jk ! dummy loop indices
748	REAL(wp) :: zcoef0, zep, cffw ! temporary scalars
749	REAL(wp) :: z1_10, cffu, cffx ! " "
750	REAL(wp) :: z1_12, cffv, cffy ! " "
751	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj
752	REAL(wp), POINTER, DIMENSION(:,:,:) :: dzx, dzy, dzz, dzu, dzv, dzw
753	REAL(wp), POINTER, DIMENSION(:,:,:) :: drhox, drhoy, drhoz, drhou, drhov, drhow
754	REAL(wp), POINTER, DIMENSION(:,:,:) :: rho_i, rho_j, rho_k
755	!!----------------------------------------------------------------------
756	!
757	CALL wrk_alloc( jpi, jpj, jpk, dzx , dzy , dzz , dzu , dzv , dzw )
758	CALL wrk_alloc( jpi, jpj, jpk, drhox, drhoy, drhoz, drhou, drhov, drhow )
759	CALL wrk_alloc( jpi, jpj, jpk, rho_i, rho_j, rho_k, zhpi, zhpj )
760	!
761
762	IF( kt == nit000 ) THEN
763	IF(lwp) WRITE(numout,*)
764	IF(lwp) WRITE(numout,*) 'dyn:hpg_djc : hydrostatic pressure gradient trend'
765	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, density Jacobian with cubic polynomial scheme'
766	ENDIF
767
768	! Local constant initialization
769	zcoef0 = - grav * 0.5_wp
770	z1_10 = 1._wp / 10._wp
771	z1_12 = 1._wp / 12._wp
772
773	!----------------------------------------------------------------------------------------
774	! compute and store in provisional arrays elementary vertical and horizontal differences
775	!----------------------------------------------------------------------------------------
776
777	!!bug gm Not a true bug, but... dzz=e3w for dzx, dzy verify what it is really
778
779	DO jk = 2, jpkm1
780	DO jj = 2, jpjm1
781	DO ji = fs_2, fs_jpim1 ! vector opt.
782	drhoz(ji,jj,jk) = rhd (ji ,jj ,jk) - rhd (ji,jj,jk-1)
783	dzz (ji,jj,jk) = fsde3w(ji ,jj ,jk) - fsde3w(ji,jj,jk-1)
784	drhox(ji,jj,jk) = rhd (ji+1,jj ,jk) - rhd (ji,jj,jk )
785	dzx (ji,jj,jk) = fsde3w(ji+1,jj ,jk) - fsde3w(ji,jj,jk )
786	drhoy(ji,jj,jk) = rhd (ji ,jj+1,jk) - rhd (ji,jj,jk )
787	dzy (ji,jj,jk) = fsde3w(ji ,jj+1,jk) - fsde3w(ji,jj,jk )
788	END DO
789	END DO
790	END DO
791
792	!-------------------------------------------------------------------------
793	! compute harmonic averages using eq. 5.18
794	!-------------------------------------------------------------------------
795	zep = 1.e-15
796
797	!!bug gm drhoz not defined at level 1 and used (jk-1 with jk=2)
798	!!bug gm idem for drhox, drhoy et ji=jpi and jj=jpj
799
800	DO jk = 2, jpkm1
801	DO jj = 2, jpjm1
802	DO ji = fs_2, fs_jpim1 ! vector opt.
803	cffw = 2._wp * drhoz(ji ,jj ,jk) * drhoz(ji,jj,jk-1)
804
805	cffu = 2._wp * drhox(ji+1,jj ,jk) * drhox(ji,jj,jk )
806	cffx = 2._wp * dzx (ji+1,jj ,jk) * dzx (ji,jj,jk )
807
808	cffv = 2._wp * drhoy(ji ,jj+1,jk) * drhoy(ji,jj,jk )
809	cffy = 2._wp * dzy (ji ,jj+1,jk) * dzy (ji,jj,jk )
810
811	IF( cffw > zep) THEN
812	drhow(ji,jj,jk) = 2._wp * drhoz(ji,jj,jk) * drhoz(ji,jj,jk-1) &
813	& / ( drhoz(ji,jj,jk) + drhoz(ji,jj,jk-1) )
814	ELSE
815	drhow(ji,jj,jk) = 0._wp
816	ENDIF
817
818	dzw(ji,jj,jk) = 2._wp * dzz(ji,jj,jk) * dzz(ji,jj,jk-1) &
819	& / ( dzz(ji,jj,jk) + dzz(ji,jj,jk-1) )
820
821	IF( cffu > zep ) THEN
822	drhou(ji,jj,jk) = 2._wp * drhox(ji+1,jj,jk) * drhox(ji,jj,jk) &
823	& / ( drhox(ji+1,jj,jk) + drhox(ji,jj,jk) )
824	ELSE
825	drhou(ji,jj,jk ) = 0._wp
826	ENDIF
827
828	IF( cffx > zep ) THEN
829	dzu(ji,jj,jk) = 2._wp * dzx(ji+1,jj,jk) * dzx(ji,jj,jk) &
830	& / ( dzx(ji+1,jj,jk) + dzx(ji,jj,jk) )
831	ELSE
832	dzu(ji,jj,jk) = 0._wp
833	ENDIF
834
835	IF( cffv > zep ) THEN
836	drhov(ji,jj,jk) = 2._wp * drhoy(ji,jj+1,jk) * drhoy(ji,jj,jk) &
837	& / ( drhoy(ji,jj+1,jk) + drhoy(ji,jj,jk) )
838	ELSE
839	drhov(ji,jj,jk) = 0._wp
840	ENDIF
841
842	IF( cffy > zep ) THEN
843	dzv(ji,jj,jk) = 2._wp * dzy(ji,jj+1,jk) * dzy(ji,jj,jk) &
844	& / ( dzy(ji,jj+1,jk) + dzy(ji,jj,jk) )
845	ELSE
846	dzv(ji,jj,jk) = 0._wp
847	ENDIF
848
849	END DO
850	END DO
851	END DO
852
853	!----------------------------------------------------------------------------------
854	! apply boundary conditions at top and bottom using 5.36-5.37
855	!----------------------------------------------------------------------------------
856	drhow(:,:, 1 ) = 1.5_wp * ( drhoz(:,:, 2 ) - drhoz(:,:, 1 ) ) - 0.5_wp * drhow(:,:, 2 )
857	drhou(:,:, 1 ) = 1.5_wp * ( drhox(:,:, 2 ) - drhox(:,:, 1 ) ) - 0.5_wp * drhou(:,:, 2 )
858	drhov(:,:, 1 ) = 1.5_wp * ( drhoy(:,:, 2 ) - drhoy(:,:, 1 ) ) - 0.5_wp * drhov(:,:, 2 )
859
860	drhow(:,:,jpk) = 1.5_wp * ( drhoz(:,:,jpk) - drhoz(:,:,jpkm1) ) - 0.5_wp * drhow(:,:,jpkm1)
861	drhou(:,:,jpk) = 1.5_wp * ( drhox(:,:,jpk) - drhox(:,:,jpkm1) ) - 0.5_wp * drhou(:,:,jpkm1)
862	drhov(:,:,jpk) = 1.5_wp * ( drhoy(:,:,jpk) - drhoy(:,:,jpkm1) ) - 0.5_wp * drhov(:,:,jpkm1)
863
864
865	!--------------------------------------------------------------
866	! Upper half of top-most grid box, compute and store
867	!-------------------------------------------------------------
868
869	!!bug gm : e3w-de3w = 0.5*e3w .... and de3w(2)-de3w(1)=e3w(2) .... to be verified
870	! true if de3w is really defined as the sum of the e3w scale factors as, it seems to me, it should be
871
872	DO jj = 2, jpjm1
873	DO ji = fs_2, fs_jpim1 ! vector opt.
874	rho_k(ji,jj,1) = -grav * ( fse3w(ji,jj,1) - fsde3w(ji,jj,1) ) &
875	& * ( rhd(ji,jj,1) &
876	& + 0.5_wp * ( rhd(ji,jj,2) - rhd(ji,jj,1) ) &
877	& * ( fse3w (ji,jj,1) - fsde3w(ji,jj,1) ) &
878	& / ( fsde3w(ji,jj,2) - fsde3w(ji,jj,1) ) )
879	END DO
880	END DO
881
882	!!bug gm : here also, simplification is possible
883	!!bug gm : optimisation: 1/10 and 1/12 the division should be done before the loop
884
885	DO jk = 2, jpkm1
886	DO jj = 2, jpjm1
887	DO ji = fs_2, fs_jpim1 ! vector opt.
888
889	rho_k(ji,jj,jk) = zcoef0 * ( rhd (ji,jj,jk) + rhd (ji,jj,jk-1) ) &
890	& * ( fsde3w(ji,jj,jk) - fsde3w(ji,jj,jk-1) ) &
891	& - grav * z1_10 * ( &
892	& ( drhow (ji,jj,jk) - drhow (ji,jj,jk-1) ) &
893	& * ( fsde3w(ji,jj,jk) - fsde3w(ji,jj,jk-1) - z1_12 * ( dzw (ji,jj,jk) + dzw (ji,jj,jk-1) ) ) &
894	& - ( dzw (ji,jj,jk) - dzw (ji,jj,jk-1) ) &
895	& * ( rhd (ji,jj,jk) - rhd (ji,jj,jk-1) - z1_12 * ( drhow(ji,jj,jk) + drhow(ji,jj,jk-1) ) ) &
896	& )
897
898	rho_i(ji,jj,jk) = zcoef0 * ( rhd (ji+1,jj,jk) + rhd (ji,jj,jk) ) &
899	& * ( fsde3w(ji+1,jj,jk) - fsde3w(ji,jj,jk) ) &
900	& - grav* z1_10 * ( &
901	& ( drhou (ji+1,jj,jk) - drhou (ji,jj,jk) ) &
902	& * ( fsde3w(ji+1,jj,jk) - fsde3w(ji,jj,jk) - z1_12 * ( dzu (ji+1,jj,jk) + dzu (ji,jj,jk) ) ) &
903	& - ( dzu (ji+1,jj,jk) - dzu (ji,jj,jk) ) &
904	& * ( rhd (ji+1,jj,jk) - rhd (ji,jj,jk) - z1_12 * ( drhou(ji+1,jj,jk) + drhou(ji,jj,jk) ) ) &
905	& )
906
907	rho_j(ji,jj,jk) = zcoef0 * ( rhd (ji,jj+1,jk) + rhd (ji,jj,jk) ) &
908	& * ( fsde3w(ji,jj+1,jk) - fsde3w(ji,jj,jk) ) &
909	& - grav* z1_10 * ( &
910	& ( drhov (ji,jj+1,jk) - drhov (ji,jj,jk) ) &
911	& * ( fsde3w(ji,jj+1,jk) - fsde3w(ji,jj,jk) - z1_12 * ( dzv (ji,jj+1,jk) + dzv (ji,jj,jk) ) ) &
912	& - ( dzv (ji,jj+1,jk) - dzv (ji,jj,jk) ) &
913	& * ( rhd (ji,jj+1,jk) - rhd (ji,jj,jk) - z1_12 * ( drhov(ji,jj+1,jk) + drhov(ji,jj,jk) ) ) &
914	& )
915
916	END DO
917	END DO
918	END DO
919	CALL lbc_lnk(rho_k,'W',1.)
920	CALL lbc_lnk(rho_i,'U',1.)
921	CALL lbc_lnk(rho_j,'V',1.)
922
923
924	! ---------------
925	! Surface value
926	! ---------------
927	DO jj = 2, jpjm1
928	DO ji = fs_2, fs_jpim1 ! vector opt.
929	zhpi(ji,jj,1) = ( rho_k(ji+1,jj ,1) - rho_k(ji,jj,1) - rho_i(ji,jj,1) ) / e1u(ji,jj)
930	zhpj(ji,jj,1) = ( rho_k(ji ,jj+1,1) - rho_k(ji,jj,1) - rho_j(ji,jj,1) ) / e2v(ji,jj)
931	! add to the general momentum trend
932	ua(ji,jj,1) = ua(ji,jj,1) + zhpi(ji,jj,1)
933	va(ji,jj,1) = va(ji,jj,1) + zhpj(ji,jj,1)
934	END DO
935	END DO
936
937	! ----------------
938	! interior value (2=<jk=<jpkm1)
939	! ----------------
940	DO jk = 2, jpkm1
941	DO jj = 2, jpjm1
942	DO ji = fs_2, fs_jpim1 ! vector opt.
943	! hydrostatic pressure gradient along s-surfaces
944	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
945	& + ( ( rho_k(ji+1,jj,jk) - rho_k(ji,jj,jk ) ) &
946	& - ( rho_i(ji ,jj,jk) - rho_i(ji,jj,jk-1) ) ) / e1u(ji,jj)
947	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
948	& + ( ( rho_k(ji,jj+1,jk) - rho_k(ji,jj,jk ) ) &
949	& -( rho_j(ji,jj ,jk) - rho_j(ji,jj,jk-1) ) ) / e2v(ji,jj)
950	! add to the general momentum trend
951	ua(ji,jj,jk) = ua(ji,jj,jk) + zhpi(ji,jj,jk)
952	va(ji,jj,jk) = va(ji,jj,jk) + zhpj(ji,jj,jk)
953	END DO
954	END DO
955	END DO
956	!
957	CALL wrk_dealloc( jpi, jpj, jpk, dzx , dzy , dzz , dzu , dzv , dzw )
958	CALL wrk_dealloc( jpi, jpj, jpk, drhox, drhoy, drhoz, drhou, drhov, drhow )
959	CALL wrk_dealloc( jpi, jpj, jpk, rho_i, rho_j, rho_k, zhpi, zhpj )
960	!
961	END SUBROUTINE hpg_djc
962
963
964	SUBROUTINE hpg_prj( kt )
965	!!---------------------------------------------------------------------
966	!! * ROUTINE hpg_prj *
967	!!
968	!! ** Method : s-coordinate case.
969	!! A Pressure-Jacobian horizontal pressure gradient method
970	!! based on the constrained cubic-spline interpolation for
971	!! all vertical coordinate systems
972	!!
973	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
974	!!----------------------------------------------------------------------
975	INTEGER, PARAMETER :: polynomial_type = 1 ! 1: cubic spline, 2: linear
976	INTEGER, INTENT(in) :: kt ! ocean time-step index
977	!!
978	INTEGER :: ji, jj, jk, jkk ! dummy loop indices
979	REAL(wp) :: zcoef0, znad ! temporary scalars
980	!!
981	!! The local variables for the correction term
982	INTEGER :: jk1, jis, jid, jjs, jjd
983	REAL(wp) :: zuijk, zvijk, zpwes, zpwed, zpnss, zpnsd, zdeps
984	REAL(wp) :: zrhdt1
985	REAL(wp) :: zdpdx1, zdpdx2, zdpdy1, zdpdy2
986	REAL(wp), POINTER, DIMENSION(:,:,:) :: zdept, zrhh
987	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zu, zv, fsp, xsp, asp, bsp, csp, dsp
988	REAL(wp), POINTER, DIMENSION(:,:) :: zsshu_n, zsshv_n
989	!!----------------------------------------------------------------------
990	!
991	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zu, zv, fsp, xsp, asp, bsp, csp, dsp )
992	CALL wrk_alloc( jpi,jpj,jpk, zdept, zrhh )
993	CALL wrk_alloc( jpi,jpj, zsshu_n, zsshv_n )
994	!
995	IF( kt == nit000 ) THEN
996	IF(lwp) WRITE(numout,*)
997	IF(lwp) WRITE(numout,*) 'dyn:hpg_prj : hydrostatic pressure gradient trend'
998	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, cubic spline pressure Jacobian'
999	ENDIF
1000
1001	!!----------------------------------------------------------------------
1002	! Local constant initialization
1003	zcoef0 = - grav
1004	znad = 0.0_wp
1005	IF( lk_vvl ) znad = 1._wp
1006
1007	! Clean 3-D work arrays
1008	zhpi(:,:,:) = 0._wp
1009	zrhh(:,:,:) = rhd(:,:,:)
1010
1011	! Preparing vertical density profile "zrhh(:,:,:)" for hybrid-sco coordinate
1012	DO jj = 1, jpj
1013	DO ji = 1, jpi
1014	jk = mbathy(ji,jj)
1015	IF( jk <= 0 ) THEN; zrhh(ji,jj,:) = 0._wp
1016	ELSE IF(jk == 1) THEN; zrhh(ji,jj, jk+1:jpk) = rhd(ji,jj,jk)
1017	ELSE IF(jk < jpkm1) THEN
1018	DO jkk = jk+1, jpk
1019	zrhh(ji,jj,jkk) = interp1(fsde3w(ji,jj,jkk), fsde3w(ji,jj,jkk-1), &
1020	fsde3w(ji,jj,jkk-2), rhd(ji,jj,jkk-1), rhd(ji,jj,jkk-2))
1021	END DO
1022	ENDIF
1023	END DO
1024	END DO
1025
1026	! Transfer the depth of "T(:,:,:)" to vertical coordinate "zdept(:,:,:)"
1027	DO jj = 1, jpj
1028	DO ji = 1, jpi
1029	zdept(ji,jj,1) = 0.5_wp * fse3w(ji,jj,1) - sshn(ji,jj) * znad
1030	END DO
1031	END DO
1032
1033	DO jk = 2, jpk
1034	DO jj = 1, jpj
1035	DO ji = 1, jpi
1036	zdept(ji,jj,jk) = zdept(ji,jj,jk-1) + fse3w(ji,jj,jk)
1037	END DO
1038	END DO
1039	END DO
1040
1041	fsp(:,:,:) = zrhh (:,:,:)
1042	xsp(:,:,:) = zdept(:,:,:)
1043
1044	! Construct the vertical density profile with the
1045	! constrained cubic spline interpolation
1046	! rho(z) = asp + bspz + cspz^2 + dsp*z^3
1047	CALL cspline(fsp,xsp,asp,bsp,csp,dsp,polynomial_type)
1048
1049	! Integrate the hydrostatic pressure "zhpi(:,:,:)" at "T(ji,jj,1)"
1050	DO jj = 2, jpj
1051	DO ji = 2, jpi
1052	zrhdt1 = zrhh(ji,jj,1) - interp3(zdept(ji,jj,1),asp(ji,jj,1), &
1053	bsp(ji,jj,1), csp(ji,jj,1), &
1054	dsp(ji,jj,1) ) * 0.25_wp * fse3w(ji,jj,1)
1055
1056	! assuming linear profile across the top half surface layer
1057	zhpi(ji,jj,1) = 0.5_wp * fse3w(ji,jj,1) * zrhdt1
1058	END DO
1059	END DO
1060
1061	! Calculate the pressure "zhpi(:,:,:)" at "T(ji,jj,2:jpkm1)"
1062	DO jk = 2, jpkm1
1063	DO jj = 2, jpj
1064	DO ji = 2, jpi
1065	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + &
1066	integ_spline(zdept(ji,jj,jk-1), zdept(ji,jj,jk),&
1067	asp(ji,jj,jk-1), bsp(ji,jj,jk-1), &
1068	csp(ji,jj,jk-1), dsp(ji,jj,jk-1))
1069	END DO
1070	END DO
1071	END DO
1072
1073	! Z coordinate of U(ji,jj,1:jpkm1) and V(ji,jj,1:jpkm1)
1074
1075	! Prepare zsshu_n and zsshv_n
1076	DO jj = 2, jpjm1
1077	DO ji = 2, jpim1
1078	zsshu_n(ji,jj) = (e12u(ji,jj) * sshn(ji,jj) + e12u(ji+1, jj) * sshn(ji+1,jj)) * &
1079	& r1_e12u(ji,jj) * umask(ji,jj,1) * 0.5_wp
1080	zsshv_n(ji,jj) = (e12v(ji,jj) * sshn(ji,jj) + e12v(ji+1, jj) * sshn(ji,jj+1)) * &
1081	& r1_e12v(ji,jj) * vmask(ji,jj,1) * 0.5_wp
1082	END DO
1083	END DO
1084
1085	DO jj = 2, jpjm1
1086	DO ji = 2, jpim1
1087	zu(ji,jj,1) = - ( fse3u(ji,jj,1) - zsshu_n(ji,jj) * znad)
1088	zv(ji,jj,1) = - ( fse3v(ji,jj,1) - zsshv_n(ji,jj) * znad)
1089	END DO
1090	END DO
1091
1092	DO jk = 2, jpkm1
1093	DO jj = 2, jpjm1
1094	DO ji = 2, jpim1
1095	zu(ji,jj,jk) = zu(ji,jj,jk-1)- fse3u(ji,jj,jk)
1096	zv(ji,jj,jk) = zv(ji,jj,jk-1)- fse3v(ji,jj,jk)
1097	END DO
1098	END DO
1099	END DO
1100
1101	DO jk = 1, jpkm1
1102	DO jj = 2, jpjm1
1103	DO ji = 2, jpim1
1104	zu(ji,jj,jk) = zu(ji,jj,jk) + 0.5_wp * fse3u(ji,jj,jk)
1105	zv(ji,jj,jk) = zv(ji,jj,jk) + 0.5_wp * fse3v(ji,jj,jk)
1106	END DO
1107	END DO
1108	END DO
1109
1110	DO jk = 1, jpkm1
1111	DO jj = 2, jpjm1
1112	DO ji = 2, jpim1
1113	zu(ji,jj,jk) = min(zu(ji,jj,jk), max(-zdept(ji,jj,jk), -zdept(ji+1,jj,jk)))
1114	zu(ji,jj,jk) = max(zu(ji,jj,jk), min(-zdept(ji,jj,jk), -zdept(ji+1,jj,jk)))
1115	zv(ji,jj,jk) = min(zv(ji,jj,jk), max(-zdept(ji,jj,jk), -zdept(ji,jj+1,jk)))
1116	zv(ji,jj,jk) = max(zv(ji,jj,jk), min(-zdept(ji,jj,jk), -zdept(ji,jj+1,jk)))
1117	END DO
1118	END DO
1119	END DO
1120
1121
1122	DO jk = 1, jpkm1
1123	DO jj = 2, jpjm1
1124	DO ji = 2, jpim1
1125	zpwes = 0._wp; zpwed = 0._wp
1126	zpnss = 0._wp; zpnsd = 0._wp
1127	zuijk = zu(ji,jj,jk)
1128	zvijk = zv(ji,jj,jk)
1129
1130	!!!!! for u equation
1131	IF( jk <= mbku(ji,jj) ) THEN
1132	IF( -zdept(ji+1,jj,jk) >= -zdept(ji,jj,jk) ) THEN
1133	jis = ji + 1; jid = ji
1134	ELSE
1135	jis = ji; jid = ji +1
1136	ENDIF
1137
1138	! integrate the pressure on the shallow side
1139	jk1 = jk
1140	DO WHILE ( -zdept(jis,jj,jk1) > zuijk )
1141	IF( jk1 == mbku(ji,jj) ) THEN
1142	zuijk = -zdept(jis,jj,jk1)
1143	EXIT
1144	ENDIF
1145	zdeps = MIN(zdept(jis,jj,jk1+1), -zuijk)
1146	zpwes = zpwes + &
1147	integ_spline(zdept(jis,jj,jk1), zdeps, &
1148	asp(jis,jj,jk1), bsp(jis,jj,jk1), &
1149	csp(jis,jj,jk1), dsp(jis,jj,jk1))
1150	jk1 = jk1 + 1
1151	END DO
1152
1153	! integrate the pressure on the deep side
1154	jk1 = jk
1155	DO WHILE ( -zdept(jid,jj,jk1) < zuijk )
1156	IF( jk1 == 1 ) THEN
1157	zdeps = zdept(jid,jj,1) + MIN(zuijk, sshn(jid,jj)*znad)
1158	zrhdt1 = zrhh(jid,jj,1) - interp3(zdept(jid,jj,1), asp(jid,jj,1), &
1159	bsp(jid,jj,1), csp(jid,jj,1), &
1160	dsp(jid,jj,1)) * zdeps
1161	zpwed = zpwed + 0.5_wp * (zrhh(jid,jj,1) + zrhdt1) * zdeps
1162	EXIT
1163	ENDIF
1164	zdeps = MAX(zdept(jid,jj,jk1-1), -zuijk)
1165	zpwed = zpwed + &
1166	integ_spline(zdeps, zdept(jid,jj,jk1), &
1167	asp(jid,jj,jk1-1), bsp(jid,jj,jk1-1), &
1168	csp(jid,jj,jk1-1), dsp(jid,jj,jk1-1) )
1169	jk1 = jk1 - 1
1170	END DO
1171
1172	! update the momentum trends in u direction
1173
1174	zdpdx1 = zcoef0 / e1u(ji,jj) * (zhpi(ji+1,jj,jk) - zhpi(ji,jj,jk))
1175	IF( lk_vvl ) THEN
1176	zdpdx2 = zcoef0 / e1u(ji,jj) * &
1177	( REAL(jis-jid, wp) * (zpwes + zpwed) + (sshn(ji+1,jj)-sshn(ji,jj)) )
1178	ELSE
1179	zdpdx2 = zcoef0 / e1u(ji,jj) * REAL(jis-jid, wp) * (zpwes + zpwed)
1180	ENDIF
1181
1182	ua(ji,jj,jk) = ua(ji,jj,jk) + (zdpdx1 + zdpdx2) * &
1183	& umask(ji,jj,jk) * tmask(ji,jj,jk) * tmask(ji+1,jj,jk)
1184	ENDIF
1185
1186	!!!!! for v equation
1187	IF( jk <= mbkv(ji,jj) ) THEN
1188	IF( -zdept(ji,jj+1,jk) >= -zdept(ji,jj,jk) ) THEN
1189	jjs = jj + 1; jjd = jj
1190	ELSE
1191	jjs = jj ; jjd = jj + 1
1192	ENDIF
1193
1194	! integrate the pressure on the shallow side
1195	jk1 = jk
1196	DO WHILE ( -zdept(ji,jjs,jk1) > zvijk )
1197	IF( jk1 == mbkv(ji,jj) ) THEN
1198	zvijk = -zdept(ji,jjs,jk1)
1199	EXIT
1200	ENDIF
1201	zdeps = MIN(zdept(ji,jjs,jk1+1), -zvijk)
1202	zpnss = zpnss + &
1203	integ_spline(zdept(ji,jjs,jk1), zdeps, &
1204	asp(ji,jjs,jk1), bsp(ji,jjs,jk1), &
1205	csp(ji,jjs,jk1), dsp(ji,jjs,jk1) )
1206	jk1 = jk1 + 1
1207	END DO
1208
1209	! integrate the pressure on the deep side
1210	jk1 = jk
1211	DO WHILE ( -zdept(ji,jjd,jk1) < zvijk )
1212	IF( jk1 == 1 ) THEN
1213	zdeps = zdept(ji,jjd,1) + MIN(zvijk, sshn(ji,jjd)*znad)
1214	zrhdt1 = zrhh(ji,jjd,1) - interp3(zdept(ji,jjd,1), asp(ji,jjd,1), &
1215	bsp(ji,jjd,1), csp(ji,jjd,1), &
1216	dsp(ji,jjd,1) ) * zdeps
1217	zpnsd = zpnsd + 0.5_wp * (zrhh(ji,jjd,1) + zrhdt1) * zdeps
1218	EXIT
1219	ENDIF
1220	zdeps = MAX(zdept(ji,jjd,jk1-1), -zvijk)
1221	zpnsd = zpnsd + &
1222	integ_spline(zdeps, zdept(ji,jjd,jk1), &
1223	asp(ji,jjd,jk1-1), bsp(ji,jjd,jk1-1), &
1224	csp(ji,jjd,jk1-1), dsp(ji,jjd,jk1-1) )
1225	jk1 = jk1 - 1
1226	END DO
1227
1228
1229	! update the momentum trends in v direction
1230
1231	zdpdy1 = zcoef0 / e2v(ji,jj) * (zhpi(ji,jj+1,jk) - zhpi(ji,jj,jk))
1232	IF( lk_vvl ) THEN
1233	zdpdy2 = zcoef0 / e2v(ji,jj) * &
1234	( REAL(jjs-jjd, wp) * (zpnss + zpnsd) + (sshn(ji,jj+1)-sshn(ji,jj)) )
1235	ELSE
1236	zdpdy2 = zcoef0 / e2v(ji,jj) * REAL(jjs-jjd, wp) * (zpnss + zpnsd )
1237	ENDIF
1238
1239	va(ji,jj,jk) = va(ji,jj,jk) + (zdpdy1 + zdpdy2)*&
1240	& vmask(ji,jj,jk)tmask(ji,jj,jk)tmask(ji,jj+1,jk)
1241	ENDIF
1242
1243
1244	END DO
1245	END DO
1246	END DO
1247	!
1248	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zu, zv, fsp, xsp, asp, bsp, csp, dsp )
1249	CALL wrk_dealloc( jpi,jpj,jpk, zdept, zrhh )
1250	CALL wrk_dealloc( jpi,jpj, zsshu_n, zsshv_n )
1251	!
1252	END SUBROUTINE hpg_prj
1253
1254
1255	SUBROUTINE cspline(fsp, xsp, asp, bsp, csp, dsp, polynomial_type)
1256	!!----------------------------------------------------------------------
1257	!! * ROUTINE cspline *
1258	!!
1259	!! ** Purpose : constrained cubic spline interpolation
1260	!!
1261	!! ** Method : f(x) = asp + bspx + cspx^2 + dsp*x^3
1262	!!
1263	!! Reference: CJC Kruger, Constrained Cubic Spline Interpoltation
1264	!!----------------------------------------------------------------------
1265	IMPLICIT NONE
1266	REAL(wp), DIMENSION(:,:,:), INTENT(in) :: fsp, xsp ! value and coordinate
1267	REAL(wp), DIMENSION(:,:,:), INTENT(out) :: asp, bsp, csp, dsp ! coefficients of
1268	! the interpoated function
1269	INTEGER, INTENT(in) :: polynomial_type ! 1: cubic spline
1270	! 2: Linear
1271	!
1272	INTEGER :: ji, jj, jk ! dummy loop indices
1273	INTEGER :: jpi, jpj, jpkm1
1274	REAL(wp) :: zdf1, zdf2, zddf1, zddf2, ztmp1, ztmp2, zdxtmp
1275	REAL(wp) :: zdxtmp1, zdxtmp2, zalpha
1276	REAL(wp) :: zdf(size(fsp,3))
1277	!!----------------------------------------------------------------------
1278
1279	jpi = size(fsp,1)
1280	jpj = size(fsp,2)
1281	jpkm1 = size(fsp,3) - 1
1282
1283
1284	IF (polynomial_type == 1) THEN ! Constrained Cubic Spline
1285	DO ji = 1, jpi
1286	DO jj = 1, jpj
1287	!!Fritsch&Butland's method, 1984 (preferred, but more computation)
1288	! DO jk = 2, jpkm1-1
1289	! zdxtmp1 = xsp(ji,jj,jk) - xsp(ji,jj,jk-1)
1290	! zdxtmp2 = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1291	! zdf1 = ( fsp(ji,jj,jk) - fsp(ji,jj,jk-1) ) / zdxtmp1
1292	! zdf2 = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp2
1293	!
1294	! zalpha = ( zdxtmp1 + 2._wp * zdxtmp2 ) / ( zdxtmp1 + zdxtmp2 ) / 3._wp
1295	!
1296	! IF(zdf1 * zdf2 <= 0._wp) THEN
1297	! zdf(jk) = 0._wp
1298	! ELSE
1299	! zdf(jk) = zdf1 * zdf2 / ( ( 1._wp - zalpha ) * zdf1 + zalpha * zdf2 )
1300	! ENDIF
1301	! END DO
1302
1303	!!Simply geometric average
1304	DO jk = 2, jpkm1-1
1305	zdf1 = (fsp(ji,jj,jk) - fsp(ji,jj,jk-1)) / (xsp(ji,jj,jk) - xsp(ji,jj,jk-1))
1306	zdf2 = (fsp(ji,jj,jk+1) - fsp(ji,jj,jk)) / (xsp(ji,jj,jk+1) - xsp(ji,jj,jk))
1307
1308	IF(zdf1 * zdf2 <= 0._wp) THEN
1309	zdf(jk) = 0._wp
1310	ELSE
1311	zdf(jk) = 2._wp * zdf1 * zdf2 / (zdf1 + zdf2)
1312	ENDIF
1313	END DO
1314
1315	zdf(1) = 1.5_wp * ( fsp(ji,jj,2) - fsp(ji,jj,1) ) / &
1316	& ( xsp(ji,jj,2) - xsp(ji,jj,1) ) - 0.5_wp * zdf(2)
1317	zdf(jpkm1) = 1.5_wp * ( fsp(ji,jj,jpkm1) - fsp(ji,jj,jpkm1-1) ) / &
1318	& ( xsp(ji,jj,jpkm1) - xsp(ji,jj,jpkm1-1) ) - &
1319	& 0.5_wp * zdf(jpkm1 - 1)
1320
1321	DO jk = 1, jpkm1 - 1
1322	zdxtmp = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1323	ztmp1 = (zdf(jk+1) + 2._wp * zdf(jk)) / zdxtmp
1324	ztmp2 = 6._wp * (fsp(ji,jj,jk+1) - fsp(ji,jj,jk)) / zdxtmp / zdxtmp
1325	zddf1 = -2._wp * ztmp1 + ztmp2
1326	ztmp1 = (2._wp * zdf(jk+1) + zdf(jk)) / zdxtmp
1327	zddf2 = 2._wp * ztmp1 - ztmp2
1328
1329	dsp(ji,jj,jk) = (zddf2 - zddf1) / 6._wp / zdxtmp
1330	csp(ji,jj,jk) = ( xsp(ji,jj,jk+1) * zddf1 - xsp(ji,jj,jk)*zddf2 ) / 2._wp / zdxtmp
1331	bsp(ji,jj,jk) = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp - &
1332	& csp(ji,jj,jk) * ( xsp(ji,jj,jk+1) + xsp(ji,jj,jk) ) - &
1333	& dsp(ji,jj,jk) * ((xsp(ji,jj,jk+1) + xsp(ji,jj,jk))**2 - &
1334	& xsp(ji,jj,jk+1) * xsp(ji,jj,jk))
1335	asp(ji,jj,jk) = fsp(ji,jj,jk) - xsp(ji,jj,jk) * (bsp(ji,jj,jk) + &
1336	& (xsp(ji,jj,jk) * (csp(ji,jj,jk) + &
1337	& dsp(ji,jj,jk) * xsp(ji,jj,jk))))
1338	END DO
1339	END DO
1340	END DO
1341
1342	ELSE IF (polynomial_type == 2) THEN ! Linear
1343	DO ji = 1, jpi
1344	DO jj = 1, jpj
1345	DO jk = 1, jpkm1-1
1346	zdxtmp =xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1347	ztmp1 = fsp(ji,jj,jk+1) - fsp(ji,jj,jk)
1348
1349	dsp(ji,jj,jk) = 0._wp
1350	csp(ji,jj,jk) = 0._wp
1351	bsp(ji,jj,jk) = ztmp1 / zdxtmp
1352	asp(ji,jj,jk) = fsp(ji,jj,jk) - bsp(ji,jj,jk) * xsp(ji,jj,jk)
1353	END DO
1354	END DO
1355	END DO
1356
1357	ELSE
1358	CALL ctl_stop( 'invalid polynomial type in cspline' )
1359	ENDIF
1360
1361	END SUBROUTINE cspline
1362
1363
1364	FUNCTION interp1(x, xl, xr, fl, fr) RESULT(f)
1365	!!----------------------------------------------------------------------
1366	!! * ROUTINE interp1 *
1367	!!
1368	!! ** Purpose : 1-d linear interpolation
1369	!!
1370	!! ** Method : interpolation is straight forward
1371	!! extrapolation is also permitted (no value limit)
1372	!!----------------------------------------------------------------------
1373	IMPLICIT NONE
1374	REAL(wp), INTENT(in) :: x, xl, xr, fl, fr
1375	REAL(wp) :: f ! result of the interpolation (extrapolation)
1376	REAL(wp) :: zdeltx
1377	!!----------------------------------------------------------------------
1378
1379	zdeltx = xr - xl
1380	IF(abs(zdeltx) <= 10._wp * EPSILON(x)) THEN
1381	f = 0.5_wp * (fl + fr)
1382	ELSE
1383	f = ( (x - xl ) * fr - ( x - xr ) * fl ) / zdeltx
1384	ENDIF
1385
1386	END FUNCTION interp1
1387
1388
1389	FUNCTION interp2(x, a, b, c, d) RESULT(f)
1390	!!----------------------------------------------------------------------
1391	!! * ROUTINE interp1 *
1392	!!
1393	!! ** Purpose : 1-d constrained cubic spline interpolation
1394	!!
1395	!! ** Method : cubic spline interpolation
1396	!!
1397	!!----------------------------------------------------------------------
1398	IMPLICIT NONE
1399	REAL(wp), INTENT(in) :: x, a, b, c, d
1400	REAL(wp) :: f ! value from the interpolation
1401	!!----------------------------------------------------------------------
1402
1403	f = a + x* ( b + x * ( c + d * x ) )
1404
1405	END FUNCTION interp2
1406
1407
1408	FUNCTION interp3(x, a, b, c, d) RESULT(f)
1409	!!----------------------------------------------------------------------
1410	!! * ROUTINE interp1 *
1411	!!
1412	!! ** Purpose : Calculate the first order of deriavtive of
1413	!! a cubic spline function y=a+bx+cx^2+d*x^3
1414	!!
1415	!! ** Method : f=dy/dx=b+2cx+3dx^2
1416	!!
1417	!!----------------------------------------------------------------------
1418	IMPLICIT NONE
1419	REAL(wp), INTENT(in) :: x, a, b, c, d
1420	REAL(wp) :: f ! value from the interpolation
1421	!!----------------------------------------------------------------------
1422
1423	f = b + x * ( 2._wp * c + 3._wp * d * x)
1424
1425	END FUNCTION interp3
1426
1427
1428	FUNCTION integ_spline(xl, xr, a, b, c, d) RESULT(f)
1429	!!----------------------------------------------------------------------
1430	!! * ROUTINE interp1 *
1431	!!
1432	!! ** Purpose : 1-d constrained cubic spline integration
1433	!!
1434	!! ** Method : integrate polynomial a+bx+cx^2+dx^3 from xl to xr
1435	!!
1436	!!----------------------------------------------------------------------
1437	IMPLICIT NONE
1438	REAL(wp), INTENT(in) :: xl, xr, a, b, c, d
1439	REAL(wp) :: za1, za2, za3
1440	REAL(wp) :: f ! integration result
1441	!!----------------------------------------------------------------------
1442
1443	za1 = 0.5_wp * b
1444	za2 = c / 3.0_wp
1445	za3 = 0.25_wp * d
1446
1447	f = xr * ( a + xr * ( za1 + xr * ( za2 + za3 * xr ) ) ) - &
1448	& xl * ( a + xl * ( za1 + xl * ( za2 + za3 * xl ) ) )
1449
1450	END FUNCTION integ_spline
1451
1452	!!======================================================================
1453	END MODULE dynhpg
1454

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