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dynhpg.F90 in branches/UKMO/dev_r5518_GO6_package_inc_asm/NEMOGCM/NEMO/OPA_SRC/DYN – NEMO

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source: branches/UKMO/dev_r5518_GO6_package_inc_asm/NEMOGCM/NEMO/OPA_SRC/DYN/dynhpg.F90 @ 8190

Last change on this file since 8190 was 8190, checked in by jwhile, 7 years ago
Update due to Tim's review
File size: 71.2 KB

Line
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 biaspar ! bias correction variables
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	REAL(wp), ALLOCATABLE, DIMENSION(:,:,:) :: z_rhd_st ! tmp density storage for pressure corr
88	REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: z_gru_st ! tmp ua trends storage for pressure corr
89	REAL(wp), ALLOCATABLE, DIMENSION(:,:) :: z_grv_st ! tmp va trends storage for pressure corr
90	!!----------------------------------------------------------------------
91	!
92	IF( nn_timing == 1 ) CALL timing_start('dyn_hpg')
93	!
94	IF( l_trddyn ) THEN ! Temporary saving of ua and va trends (l_trddyn)
95	CALL wrk_alloc( jpi,jpj,jpk, ztrdu, ztrdv )
96	ztrdu(:,:,:) = ua(:,:,:)
97	ztrdv(:,:,:) = va(:,:,:)
98	ENDIF
99	!
100	IF ( ln_bias .AND. ln_bias_pc_app ) THEN
101
102	!Allocate space for tempory variables
103	ALLOCATE( z_rhd_st(jpi,jpj,jpk), &
104	& z_gru_st(jpi,jpj), &
105	& z_grv_st(jpi,jpj) )
106
107	z_rhd_st(:,:,:) = rhd(:,:,:) ! store orig density
108	rhd(:,:,:) = rhd_pc(:,:,:) ! use pressure corrected density
109	z_gru_st(:,:) = gru(:,:)
110	gru(:,:) = gru_pc(:,:)
111	z_grv_st(:,:) = grv(:,:)
112	grv(:,:) = grv_pc(:,:)
113
114	ENDIF
115	!
116	SELECT CASE ( nhpg ) ! Hydrostatic pressure gradient computation
117	CASE ( 0 ) ; CALL hpg_zco ( kt ) ! z-coordinate
118	CASE ( 1 ) ; CALL hpg_zps ( kt ) ! z-coordinate plus partial steps (interpolation)
119	CASE ( 2 ) ; CALL hpg_sco ( kt ) ! s-coordinate (standard jacobian formulation)
120	CASE ( 3 ) ; CALL hpg_djc ( kt ) ! s-coordinate (Density Jacobian with Cubic polynomial)
121	CASE ( 4 ) ; CALL hpg_prj ( kt ) ! s-coordinate (Pressure Jacobian scheme)
122	CASE ( 5 ) ; CALL hpg_isf ( kt ) ! s-coordinate similar to sco modify for ice shelf
123	END SELECT
124	!
125	IF( l_trddyn ) THEN ! save the hydrostatic pressure gradient trends for momentum trend diagnostics
126	ztrdu(:,:,:) = ua(:,:,:) - ztrdu(:,:,:)
127	ztrdv(:,:,:) = va(:,:,:) - ztrdv(:,:,:)
128	CALL trd_dyn( ztrdu, ztrdv, jpdyn_hpg, kt )
129	CALL wrk_dealloc( jpi,jpj,jpk, ztrdu, ztrdv )
130	ENDIF
131	!
132	IF(ln_ctl) CALL prt_ctl( tab3d_1=ua, clinfo1=' hpg - Ua: ', mask1=umask, &
133	& tab3d_2=va, clinfo2= ' Va: ', mask2=vmask, clinfo3='dyn' )
134	!
135	IF ( ln_bias .AND. ln_bias_pc_app ) THEN
136	rhd(:,:,:) = z_rhd_st(:,:,:) ! restore original density
137	gru(:,:) = z_gru_st(:,:)
138	grv(:,:) = z_grv_st(:,:)
139
140	!Deallocate tempory variables
141	DEALLOCATE( z_rhd_st(jpi,jpj,jpk), &
142	& z_gru_st(jpi,jpj), &
143	& z_grv_st(jpi,jpj) )
144	ENDIF
145	!
146	IF( nn_timing == 1 ) CALL timing_stop('dyn_hpg')
147	!
148	END SUBROUTINE dyn_hpg
149
150
151	SUBROUTINE dyn_hpg_init
152	!!----------------------------------------------------------------------
153	!! * ROUTINE dyn_hpg_init *
154	!!
155	!! ** Purpose : initializations for the hydrostatic pressure gradient
156	!! computation and consistency control
157	!!
158	!! ** Action : Read the namelist namdyn_hpg and check the consistency
159	!! with the type of vertical coordinate used (zco, zps, sco)
160	!!----------------------------------------------------------------------
161	INTEGER :: ioptio = 0 ! temporary integer
162	INTEGER :: ios ! Local integer output status for namelist read
163	!!
164	NAMELIST/namdyn_hpg/ ln_hpg_zco, ln_hpg_zps, ln_hpg_sco, &
165	& ln_hpg_djc, ln_hpg_prj, ln_hpg_isf, ln_dynhpg_imp
166	!!----------------------------------------------------------------------
167	!
168	REWIND( numnam_ref ) ! Namelist namdyn_hpg in reference namelist : Hydrostatic pressure gradient
169	READ ( numnam_ref, namdyn_hpg, IOSTAT = ios, ERR = 901)
170	901 IF( ios /= 0 ) CALL ctl_nam ( ios , 'namdyn_hpg in reference namelist', lwp )
171
172	REWIND( numnam_cfg ) ! Namelist namdyn_hpg in configuration namelist : Hydrostatic pressure gradient
173	READ ( numnam_cfg, namdyn_hpg, IOSTAT = ios, ERR = 902 )
174	902 IF( ios /= 0 ) CALL ctl_nam ( ios , 'namdyn_hpg in configuration namelist', lwp )
175	IF(lwm) WRITE ( numond, namdyn_hpg )
176	!
177	IF(lwp) THEN ! Control print
178	WRITE(numout,*)
179	WRITE(numout,*) 'dyn_hpg_init : hydrostatic pressure gradient initialisation'
180	WRITE(numout,*) '~~~~~~~~~~~~'
181	WRITE(numout,*) ' Namelist namdyn_hpg : choice of hpg scheme'
182	WRITE(numout,*) ' z-coord. - full steps ln_hpg_zco = ', ln_hpg_zco
183	WRITE(numout,*) ' z-coord. - partial steps (interpolation) ln_hpg_zps = ', ln_hpg_zps
184	WRITE(numout,*) ' s-coord. (standard jacobian formulation) ln_hpg_sco = ', ln_hpg_sco
185	WRITE(numout,*) ' s-coord. (standard jacobian formulation) for isf ln_hpg_isf = ', ln_hpg_isf
186	WRITE(numout,*) ' s-coord. (Density Jacobian: Cubic polynomial) ln_hpg_djc = ', ln_hpg_djc
187	WRITE(numout,*) ' s-coord. (Pressure Jacobian: Cubic polynomial) ln_hpg_prj = ', ln_hpg_prj
188	WRITE(numout,*) ' time stepping: centered (F) or semi-implicit (T) ln_dynhpg_imp = ', ln_dynhpg_imp
189	ENDIF
190	!
191	IF( ln_hpg_djc ) &
192	& CALL ctl_stop('dyn_hpg_init : Density Jacobian: Cubic polynominal method &
193	& currently disabled (bugs under investigation). Please select &
194	& either ln_hpg_sco or ln_hpg_prj instead')
195	!
196	IF( lk_vvl .AND. .NOT. (ln_hpg_sco.OR.ln_hpg_prj.OR.ln_hpg_isf) ) &
197	& CALL ctl_stop('dyn_hpg_init : variable volume key_vvl requires:&
198	& the standard jacobian formulation hpg_sco or &
199	& the pressure jacobian formulation hpg_prj')
200
201	IF( ln_hpg_isf .AND. .NOT. ln_isfcav ) &
202	& CALL ctl_stop( ' hpg_isf not available if ln_isfcav = false ' )
203	IF( .NOT. ln_hpg_isf .AND. ln_isfcav ) &
204	& CALL ctl_stop( 'Only hpg_isf has been corrected to work with ice shelf cavity.' )
205	!
206	! ! Set nhpg from ln_hpg_... flags
207	IF( ln_hpg_zco ) nhpg = 0
208	IF( ln_hpg_zps ) nhpg = 1
209	IF( ln_hpg_sco ) nhpg = 2
210	IF( ln_hpg_djc ) nhpg = 3
211	IF( ln_hpg_prj ) nhpg = 4
212	IF( ln_hpg_isf ) nhpg = 5
213	!
214	! ! Consistency check
215	ioptio = 0
216	IF( ln_hpg_zco ) ioptio = ioptio + 1
217	IF( ln_hpg_zps ) ioptio = ioptio + 1
218	IF( ln_hpg_sco ) ioptio = ioptio + 1
219	IF( ln_hpg_djc ) ioptio = ioptio + 1
220	IF( ln_hpg_prj ) ioptio = ioptio + 1
221	IF( ln_hpg_isf ) ioptio = ioptio + 1
222	IF( ioptio /= 1 ) CALL ctl_stop( 'NO or several hydrostatic pressure gradient options used' )
223	!
224	! initialisation of ice load
225	riceload(:,:)=0.0
226	!
227	END SUBROUTINE dyn_hpg_init
228
229
230	SUBROUTINE hpg_zco( kt )
231	!!---------------------------------------------------------------------
232	!! * ROUTINE hpg_zco *
233	!!
234	!! ** Method : z-coordinate case, levels are horizontal surfaces.
235	!! The now hydrostatic pressure gradient at a given level, jk,
236	!! is computed by taking the vertical integral of the in-situ
237	!! density gradient along the model level from the suface to that
238	!! level: zhpi = grav .....
239	!! zhpj = grav .....
240	!! add it to the general momentum trend (ua,va).
241	!! ua = ua - 1/e1u * zhpi
242	!! va = va - 1/e2v * zhpj
243	!!
244	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
245	!!----------------------------------------------------------------------
246	INTEGER, INTENT(in) :: kt ! ocean time-step index
247	!!
248	INTEGER :: ji, jj, jk ! dummy loop indices
249	REAL(wp) :: zcoef0, zcoef1 ! temporary scalars
250	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj
251	!!----------------------------------------------------------------------
252	!
253	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zhpj )
254	!
255	IF( kt == nit000 ) THEN
256	IF(lwp) WRITE(numout,*)
257	IF(lwp) WRITE(numout,*) 'dyn:hpg_zco : hydrostatic pressure gradient trend'
258	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ z-coordinate case '
259	ENDIF
260
261	zcoef0 = - grav * 0.5_wp ! Local constant initialization
262
263	! Surface value
264	DO jj = 2, jpjm1
265	DO ji = fs_2, fs_jpim1 ! vector opt.
266	zcoef1 = zcoef0 * fse3w(ji,jj,1)
267	! hydrostatic pressure gradient
268	zhpi(ji,jj,1) = zcoef1 * ( rhd(ji+1,jj,1) - rhd(ji,jj,1) ) / e1u(ji,jj)
269	zhpj(ji,jj,1) = zcoef1 * ( rhd(ji,jj+1,1) - rhd(ji,jj,1) ) / e2v(ji,jj)
270	! add to the general momentum trend
271	ua(ji,jj,1) = ua(ji,jj,1) + zhpi(ji,jj,1)
272	va(ji,jj,1) = va(ji,jj,1) + zhpj(ji,jj,1)
273	END DO
274	END DO
275
276	!
277	! interior value (2=<jk=<jpkm1)
278	DO jk = 2, jpkm1
279	DO jj = 2, jpjm1
280	DO ji = fs_2, fs_jpim1 ! vector opt.
281	zcoef1 = zcoef0 * fse3w(ji,jj,jk)
282	! hydrostatic pressure gradient
283	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
284	& + zcoef1 * ( ( rhd(ji+1,jj,jk)+rhd(ji+1,jj,jk-1) ) &
285	& - ( rhd(ji ,jj,jk)+rhd(ji ,jj,jk-1) ) ) / e1u(ji,jj)
286
287	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
288	& + zcoef1 * ( ( rhd(ji,jj+1,jk)+rhd(ji,jj+1,jk-1) ) &
289	& - ( rhd(ji,jj, jk)+rhd(ji,jj ,jk-1) ) ) / e2v(ji,jj)
290	! add to the general momentum trend
291	ua(ji,jj,jk) = ua(ji,jj,jk) + zhpi(ji,jj,jk)
292	va(ji,jj,jk) = va(ji,jj,jk) + zhpj(ji,jj,jk)
293	END DO
294	END DO
295	END DO
296	!
297	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zhpj )
298	!
299	END SUBROUTINE hpg_zco
300
301
302	SUBROUTINE hpg_zps( kt )
303	!!---------------------------------------------------------------------
304	!! * ROUTINE hpg_zps *
305	!!
306	!! ** Method : z-coordinate plus partial steps case. blahblah...
307	!!
308	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
309	!!----------------------------------------------------------------------
310	INTEGER, INTENT(in) :: kt ! ocean time-step index
311	!!
312	INTEGER :: ji, jj, jk ! dummy loop indices
313	INTEGER :: iku, ikv ! temporary integers
314	REAL(wp) :: zcoef0, zcoef1, zcoef2, zcoef3 ! temporary scalars
315	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj
316	!!----------------------------------------------------------------------
317	!
318	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zhpj )
319	!
320	IF( kt == nit000 ) THEN
321	IF(lwp) WRITE(numout,*)
322	IF(lwp) WRITE(numout,*) 'dyn:hpg_zps : hydrostatic pressure gradient trend'
323	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ z-coordinate with partial steps - vector optimization'
324	ENDIF
325
326
327	! Local constant initialization
328	zcoef0 = - grav * 0.5_wp
329
330	! Surface value (also valid in partial step case)
331	DO jj = 2, jpjm1
332	DO ji = fs_2, fs_jpim1 ! vector opt.
333	zcoef1 = zcoef0 * fse3w(ji,jj,1)
334	! hydrostatic pressure gradient
335	zhpi(ji,jj,1) = zcoef1 * ( rhd(ji+1,jj ,1) - rhd(ji,jj,1) ) / e1u(ji,jj)
336	zhpj(ji,jj,1) = zcoef1 * ( rhd(ji ,jj+1,1) - rhd(ji,jj,1) ) / e2v(ji,jj)
337	! add to the general momentum trend
338	ua(ji,jj,1) = ua(ji,jj,1) + zhpi(ji,jj,1)
339	va(ji,jj,1) = va(ji,jj,1) + zhpj(ji,jj,1)
340	END DO
341	END DO
342
343
344	! interior value (2=<jk=<jpkm1)
345	DO jk = 2, jpkm1
346	DO jj = 2, jpjm1
347	DO ji = fs_2, fs_jpim1 ! vector opt.
348	zcoef1 = zcoef0 * fse3w(ji,jj,jk)
349	! hydrostatic pressure gradient
350	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
351	& + zcoef1 * ( ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) ) &
352	& - ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) ) ) / e1u(ji,jj)
353
354	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
355	& + zcoef1 * ( ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) ) &
356	& - ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) ) ) / e2v(ji,jj)
357	! add to the general momentum trend
358	ua(ji,jj,jk) = ua(ji,jj,jk) + zhpi(ji,jj,jk)
359	va(ji,jj,jk) = va(ji,jj,jk) + zhpj(ji,jj,jk)
360	END DO
361	END DO
362	END DO
363
364
365	! partial steps correction at the last level (use gru & grv computed in zpshde.F90)
366	DO jj = 2, jpjm1
367	DO ji = 2, jpim1
368	iku = mbku(ji,jj)
369	ikv = mbkv(ji,jj)
370	zcoef2 = zcoef0 * MIN( fse3w(ji,jj,iku), fse3w(ji+1,jj ,iku) )
371	zcoef3 = zcoef0 * MIN( fse3w(ji,jj,ikv), fse3w(ji ,jj+1,ikv) )
372	IF( iku > 1 ) THEN ! on i-direction (level 2 or more)
373	ua (ji,jj,iku) = ua(ji,jj,iku) - zhpi(ji,jj,iku) ! subtract old value
374	zhpi(ji,jj,iku) = zhpi(ji,jj,iku-1) & ! compute the new one
375	& + zcoef2 * ( rhd(ji+1,jj,iku-1) - rhd(ji,jj,iku-1) + gru(ji,jj) ) / e1u(ji,jj)
376	ua (ji,jj,iku) = ua(ji,jj,iku) + zhpi(ji,jj,iku) ! add the new one to the general momentum trend
377	ENDIF
378	IF( ikv > 1 ) THEN ! on j-direction (level 2 or more)
379	va (ji,jj,ikv) = va(ji,jj,ikv) - zhpj(ji,jj,ikv) ! subtract old value
380	zhpj(ji,jj,ikv) = zhpj(ji,jj,ikv-1) & ! compute the new one
381	& + zcoef3 * ( rhd(ji,jj+1,ikv-1) - rhd(ji,jj,ikv-1) + grv(ji,jj) ) / e2v(ji,jj)
382	va (ji,jj,ikv) = va(ji,jj,ikv) + zhpj(ji,jj,ikv) ! add the new one to the general momentum trend
383	ENDIF
384	END DO
385	END DO
386	!
387	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zhpj )
388	!
389	END SUBROUTINE hpg_zps
390
391	SUBROUTINE hpg_sco( kt )
392	!!---------------------------------------------------------------------
393	!! * ROUTINE hpg_sco *
394	!!
395	!! ** Method : s-coordinate case. Jacobian scheme.
396	!! The now hydrostatic pressure gradient at a given level, jk,
397	!! is computed by taking the vertical integral of the in-situ
398	!! density gradient along the model level from the suface to that
399	!! level. s-coordinates (ln_sco): a corrective term is added
400	!! to the horizontal pressure gradient :
401	!! zhpi = grav ..... + 1/e1u mi(rhd) di[ grav dep3w ]
402	!! zhpj = grav ..... + 1/e2v mj(rhd) dj[ grav dep3w ]
403	!! add it to the general momentum trend (ua,va).
404	!! ua = ua - 1/e1u * zhpi
405	!! va = va - 1/e2v * zhpj
406	!!
407	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
408	!!----------------------------------------------------------------------
409	INTEGER, INTENT(in) :: kt ! ocean time-step index
410	!!
411	INTEGER :: ji, jj, jk ! dummy loop indices
412	REAL(wp) :: zcoef0, zuap, zvap, znad ! temporary scalars
413	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj
414	!!----------------------------------------------------------------------
415	!
416	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zhpj )
417	!
418	IF( kt == nit000 ) THEN
419	IF(lwp) WRITE(numout,*)
420	IF(lwp) WRITE(numout,*) 'dyn:hpg_sco : hydrostatic pressure gradient trend'
421	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, OPA original scheme used'
422	ENDIF
423
424	! Local constant initialization
425	zcoef0 = - grav * 0.5_wp
426	! To use density and not density anomaly
427	IF ( lk_vvl ) THEN ; znad = 1._wp ! Variable volume
428	ELSE ; znad = 0._wp ! Fixed volume
429	ENDIF
430
431	! Surface value
432	DO jj = 2, jpjm1
433	DO ji = fs_2, fs_jpim1 ! vector opt.
434	! hydrostatic pressure gradient along s-surfaces
435	zhpi(ji,jj,1) = zcoef0 / e1u(ji,jj) * ( fse3w(ji+1,jj ,1) * ( znad + rhd(ji+1,jj ,1) ) &
436	& - fse3w(ji ,jj ,1) * ( znad + rhd(ji ,jj ,1) ) )
437	zhpj(ji,jj,1) = zcoef0 / e2v(ji,jj) * ( fse3w(ji ,jj+1,1) * ( znad + rhd(ji ,jj+1,1) ) &
438	& - fse3w(ji ,jj ,1) * ( znad + rhd(ji ,jj ,1) ) )
439	! s-coordinate pressure gradient correction
440	zuap = -zcoef0 * ( rhd (ji+1,jj,1) + rhd (ji,jj,1) + 2._wp * znad ) &
441	& * ( fsde3w(ji+1,jj,1) - fsde3w(ji,jj,1) ) / e1u(ji,jj)
442	zvap = -zcoef0 * ( rhd (ji,jj+1,1) + rhd (ji,jj,1) + 2._wp * znad ) &
443	& * ( fsde3w(ji,jj+1,1) - fsde3w(ji,jj,1) ) / e2v(ji,jj)
444	! add to the general momentum trend
445	ua(ji,jj,1) = ua(ji,jj,1) + zhpi(ji,jj,1) + zuap
446	va(ji,jj,1) = va(ji,jj,1) + zhpj(ji,jj,1) + zvap
447	END DO
448	END DO
449
450	! interior value (2=<jk=<jpkm1)
451	DO jk = 2, jpkm1
452	DO jj = 2, jpjm1
453	DO ji = fs_2, fs_jpim1 ! vector opt.
454	! hydrostatic pressure gradient along s-surfaces
455	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + zcoef0 / e1u(ji,jj) &
456	& * ( fse3w(ji+1,jj,jk) * ( rhd(ji+1,jj,jk) + rhd(ji+1,jj,jk-1) + 2*znad ) &
457	& - fse3w(ji ,jj,jk) * ( rhd(ji ,jj,jk) + rhd(ji ,jj,jk-1) + 2*znad ) )
458	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) + zcoef0 / e2v(ji,jj) &
459	& * ( fse3w(ji,jj+1,jk) * ( rhd(ji,jj+1,jk) + rhd(ji,jj+1,jk-1) + 2*znad ) &
460	& - fse3w(ji,jj ,jk) * ( rhd(ji,jj, jk) + rhd(ji,jj ,jk-1) + 2*znad ) )
461	! s-coordinate pressure gradient correction
462	zuap = -zcoef0 * ( rhd (ji+1,jj ,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
463	& * ( fsde3w(ji+1,jj ,jk) - fsde3w(ji,jj,jk) ) / e1u(ji,jj)
464	zvap = -zcoef0 * ( rhd (ji ,jj+1,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
465	& * ( fsde3w(ji ,jj+1,jk) - fsde3w(ji,jj,jk) ) / e2v(ji,jj)
466	! add to the general momentum trend
467	ua(ji,jj,jk) = ua(ji,jj,jk) + zhpi(ji,jj,jk) + zuap
468	va(ji,jj,jk) = va(ji,jj,jk) + zhpj(ji,jj,jk) + zvap
469	END DO
470	END DO
471	END DO
472	!
473	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zhpj )
474	!
475	END SUBROUTINE hpg_sco
476
477	SUBROUTINE hpg_isf( kt )
478	!!---------------------------------------------------------------------
479	!! * ROUTINE hpg_sco *
480	!!
481	!! ** Method : s-coordinate case. Jacobian scheme.
482	!! The now hydrostatic pressure gradient at a given level, jk,
483	!! is computed by taking the vertical integral of the in-situ
484	!! density gradient along the model level from the suface to that
485	!! level. s-coordinates (ln_sco): a corrective term is added
486	!! to the horizontal pressure gradient :
487	!! zhpi = grav ..... + 1/e1u mi(rhd) di[ grav dep3w ]
488	!! zhpj = grav ..... + 1/e2v mj(rhd) dj[ grav dep3w ]
489	!! add it to the general momentum trend (ua,va).
490	!! ua = ua - 1/e1u * zhpi
491	!! va = va - 1/e2v * zhpj
492	!! iceload is added and partial cell case are added to the top and bottom
493	!!
494	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
495	!!----------------------------------------------------------------------
496	INTEGER, INTENT(in) :: kt ! ocean time-step index
497	!!
498	INTEGER :: ji, jj, jk, iku, ikv, ikt, iktp1i, iktp1j ! dummy loop indices
499	REAL(wp) :: zcoef0, zuap, zvap, znad, ze3wu, ze3wv, zuapint, zvapint, zhpjint, zhpiint, zdzwt, zdzwtjp1, zdzwtip1 ! temporary scalars
500	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj, zrhd
501	REAL(wp), POINTER, DIMENSION(:,:,:) :: ztstop
502	REAL(wp), POINTER, DIMENSION(:,:) :: ze3w, zp, zrhdtop_isf, zrhdtop_oce, ziceload, zdept, zpshpi, zpshpj
503	!!----------------------------------------------------------------------
504	!
505	CALL wrk_alloc( jpi,jpj, 2, ztstop)
506	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zhpj, zrhd)
507	CALL wrk_alloc( jpi,jpj, ze3w, zp, zrhdtop_isf, zrhdtop_oce, ziceload, zdept, zpshpi, zpshpj)
508	!
509	IF( kt == nit000 ) THEN
510	IF(lwp) WRITE(numout,*)
511	IF(lwp) WRITE(numout,*) 'dyn:hpg_isf : hydrostatic pressure gradient trend for ice shelf'
512	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, OPA original scheme used'
513	ENDIF
514
515	! Local constant initialization
516	zcoef0 = - grav * 0.5_wp
517	! To use density and not density anomaly
518	! IF ( lk_vvl ) THEN ; znad = 1._wp ! Variable volume
519	! ELSE ; znad = 0._wp ! Fixed volume
520	! ENDIF
521	znad=1._wp
522	! iniitialised to 0. zhpi zhpi
523	zhpi(:,:,:)=0._wp ; zhpj(:,:,:)=0._wp
524
525	!==================================================================================
526	!=====Compute iceload and contribution of the half first wet layer =================
527	!===================================================================================
528
529	! assume water displaced by the ice shelf is at T=-1.9 and S=34.4 (rude)
530	ztstop(:,:,1)=-1.9_wp ; ztstop(:,:,2)=34.4_wp
531
532	! compute density of the water displaced by the ice shelf
533	zrhd = rhd ! save rhd
534	DO jk = 1, jpk
535	zdept(:,:)=gdept_1d(jk)
536	CALL eos(ztstop(:,:,:),zdept(:,:),rhd(:,:,jk))
537	END DO
538	WHERE ( tmask(:,:,:) == 1._wp)
539	rhd(:,:,:) = zrhd(:,:,:) ! replace wet cell by the saved rhd
540	END WHERE
541
542	! compute rhd at the ice/oce interface (ice shelf side)
543	CALL eos(ztstop,risfdep,zrhdtop_isf)
544
545	! compute rhd at the ice/oce interface (ocean side)
546	DO ji=1,jpi
547	DO jj=1,jpj
548	ikt=mikt(ji,jj)
549	ztstop(ji,jj,1)=tsn(ji,jj,ikt,1)
550	ztstop(ji,jj,2)=tsn(ji,jj,ikt,2)
551	END DO
552	END DO
553	CALL eos(ztstop,risfdep,zrhdtop_oce)
554	!
555	! Surface value + ice shelf gradient
556	! compute pressure due to ice shelf load (used to compute hpgi/j for all the level from 1 to miku/v)
557	ziceload = 0._wp
558	DO jj = 1, jpj
559	DO ji = 1, jpi ! vector opt.
560	ikt=mikt(ji,jj)
561	ziceload(ji,jj) = ziceload(ji,jj) + (znad + rhd(ji,jj,1) ) * fse3w(ji,jj,1) * (1._wp - tmask(ji,jj,1))
562	DO jk=2,ikt-1
563	ziceload(ji,jj) = ziceload(ji,jj) + (2._wp * znad + rhd(ji,jj,jk-1) + rhd(ji,jj,jk)) * fse3w(ji,jj,jk) &
564	& * (1._wp - tmask(ji,jj,jk))
565	END DO
566	IF (ikt .GE. 2) ziceload(ji,jj) = ziceload(ji,jj) + (2._wp * znad + zrhdtop_isf(ji,jj) + rhd(ji,jj,ikt-1)) &
567	& * ( risfdep(ji,jj) - gdept_1d(ikt-1) )
568	END DO
569	END DO
570	riceload(:,:) = 0.0_wp ; riceload(:,:)=ziceload(:,:) ! need to be saved for diaar5
571	! compute zp from z=0 to first T wet point (correction due to zps not yet applied)
572	DO jj = 2, jpjm1
573	DO ji = fs_2, fs_jpim1 ! vector opt.
574	ikt=mikt(ji,jj) ; iktp1i=mikt(ji+1,jj); iktp1j=mikt(ji,jj+1)
575	! hydrostatic pressure gradient along s-surfaces and ice shelf pressure
576	! we assume ISF is in isostatic equilibrium
577	zhpi(ji,jj,1) = zcoef0 / e1u(ji,jj) * ( 0.5_wp * fse3w(ji+1,jj ,iktp1i) &
578	& * ( 2._wp * znad + rhd(ji+1,jj ,iktp1i) + zrhdtop_oce(ji+1,jj ) ) &
579	& - 0.5_wp * fse3w(ji ,jj ,ikt ) &
580	& * ( 2._wp * znad + rhd(ji ,jj ,ikt ) + zrhdtop_oce(ji ,jj ) ) &
581	& + ( ziceload(ji+1,jj) - ziceload(ji,jj)) )
582	zhpj(ji,jj,1) = zcoef0 / e2v(ji,jj) * ( 0.5_wp * fse3w(ji ,jj+1,iktp1j) &
583	& * ( 2._wp * znad + rhd(ji ,jj+1,iktp1j) + zrhdtop_oce(ji ,jj+1) ) &
584	& - 0.5_wp * fse3w(ji ,jj ,ikt ) &
585	& * ( 2._wp * znad + rhd(ji ,jj ,ikt ) + zrhdtop_oce(ji ,jj ) ) &
586	& + ( ziceload(ji,jj+1) - ziceload(ji,jj) ) )
587	! s-coordinate pressure gradient correction (=0 if z coordinate)
588	zuap = -zcoef0 * ( rhd (ji+1,jj,1) + rhd (ji,jj,1) + 2._wp * znad ) &
589	& * ( fsde3w(ji+1,jj,1) - fsde3w(ji,jj,1) ) / e1u(ji,jj)
590	zvap = -zcoef0 * ( rhd (ji,jj+1,1) + rhd (ji,jj,1) + 2._wp * znad ) &
591	& * ( fsde3w(ji,jj+1,1) - fsde3w(ji,jj,1) ) / e2v(ji,jj)
592	! add to the general momentum trend
593	ua(ji,jj,1) = ua(ji,jj,1) + (zhpi(ji,jj,1) + zuap) * umask(ji,jj,1)
594	va(ji,jj,1) = va(ji,jj,1) + (zhpj(ji,jj,1) + zvap) * vmask(ji,jj,1)
595	END DO
596	END DO
597	!==================================================================================
598	!===== Compute partial cell contribution for the top cell =========================
599	!==================================================================================
600	DO jj = 2, jpjm1
601	DO ji = fs_2, fs_jpim1 ! vector opt.
602	iku = miku(ji,jj) ;
603	zpshpi(ji,jj)=0.0_wp ; zpshpj(ji,jj)=0.0_wp
604	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))
605	! u direction
606	IF ( iku .GT. 1 ) THEN
607	! case iku
608	zhpi(ji,jj,iku) = zcoef0 / e1u(ji,jj) * ze3wu &
609	& * ( rhd (ji+1,jj,iku) + rhd (ji,jj,iku) &
610	& + SIGN(1._wp,ze3wu) * grui(ji,jj) + 2._wp * znad )
611	! corrective term ( = 0 if z coordinate )
612	zuap = -zcoef0 * ( arui(ji,jj) + 2._wp * znad ) * gzui(ji,jj) / e1u(ji,jj)
613	! zhpi will be added in interior loop
614	ua(ji,jj,iku) = ua(ji,jj,iku) + zuap
615	! in case of 2 cell water column, need to save the pressure gradient to compute the bottom pressure
616	IF (mbku(ji,jj) == iku + 1) zpshpi(ji,jj) = zhpi(ji,jj,iku)
617
618	! case iku + 1 (remove the zphi term added in the interior loop and compute the one corrected for zps)
619	zhpiint = zcoef0 / e1u(ji,jj) &
620	& * ( fse3w(ji+1,jj ,iku+1) * ( (rhd(ji+1,jj,iku+1) + znad) &
621	& + (rhd(ji+1,jj,iku ) + znad) ) * tmask(ji+1,jj,iku) &
622	& - fse3w(ji ,jj ,iku+1) * ( (rhd(ji ,jj,iku+1) + znad) &
623	& + (rhd(ji ,jj,iku ) + znad) ) * tmask(ji ,jj,iku) )
624	zhpi(ji,jj,iku+1) = zcoef0 / e1u(ji,jj) * ge3rui(ji,jj) - zhpiint
625	END IF
626
627	! v direction
628	ikv = mikv(ji,jj)
629	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))
630	IF ( ikv .GT. 1 ) THEN
631	! case ikv
632	zhpj(ji,jj,ikv) = zcoef0 / e2v(ji,jj) * ze3wv &
633	& * ( rhd(ji,jj+1,ikv) + rhd (ji,jj,ikv) &
634	& + SIGN(1._wp,ze3wv) * grvi(ji,jj) + 2._wp * znad )
635	! corrective term ( = 0 if z coordinate )
636	zvap = -zcoef0 * ( arvi(ji,jj) + 2._wp * znad ) * gzvi(ji,jj) / e2v(ji,jj)
637	! zhpi will be added in interior loop
638	va(ji,jj,ikv) = va(ji,jj,ikv) + zvap
639	! in case of 2 cell water column, need to save the pressure gradient to compute the bottom pressure
640	IF (mbkv(ji,jj) == ikv + 1) zpshpj(ji,jj) = zhpj(ji,jj,ikv)
641
642	! case ikv + 1 (remove the zphj term added in the interior loop and compute the one corrected for zps)
643	zhpjint = zcoef0 / e2v(ji,jj) &
644	& * ( fse3w(ji ,jj+1,ikv+1) * ( (rhd(ji,jj+1,ikv+1) + znad) &
645	& + (rhd(ji,jj+1,ikv ) + znad) ) * tmask(ji,jj+1,ikv) &
646	& - fse3w(ji ,jj ,ikv+1) * ( (rhd(ji,jj ,ikv+1) + znad) &
647	& + (rhd(ji,jj ,ikv ) + znad) ) * tmask(ji,jj ,ikv) )
648	zhpj(ji,jj,ikv+1) = zcoef0 / e2v(ji,jj) * ge3rvi(ji,jj) - zhpjint
649	END IF
650	END DO
651	END DO
652
653	!==================================================================================
654	!===== Compute interior value =====================================================
655	!==================================================================================
656
657	DO jj = 2, jpjm1
658	DO ji = fs_2, fs_jpim1 ! vector opt.
659	iku=miku(ji,jj); ikv=mikv(ji,jj)
660	DO jk = 2, jpkm1
661	! hydrostatic pressure gradient along s-surfaces
662	! zhpi is masked for the first wet cell (contribution already done in the upper bloc)
663	zhpi(ji,jj,jk) = zhpi(ji,jj,jk) + zhpi(ji,jj,jk-1) &
664	& + zcoef0 / e1u(ji,jj) &
665	& * ( fse3w(ji+1,jj ,jk) * ( (rhd(ji+1,jj,jk ) + znad) &
666	& + (rhd(ji+1,jj,jk-1) + znad) ) * tmask(ji+1,jj,jk-1) &
667	& - fse3w(ji ,jj ,jk) * ( (rhd(ji ,jj,jk ) + znad) &
668	& + (rhd(ji ,jj,jk-1) + znad) ) * tmask(ji ,jj,jk-1) )
669	! s-coordinate pressure gradient correction
670	! corrective term, we mask this term for the first wet level beneath the ice shelf (contribution done in the upper bloc)
671	zuap = - zcoef0 * ( rhd (ji+1,jj ,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
672	& * ( fsde3w(ji+1,jj ,jk) - fsde3w(ji,jj,jk) ) / e1u(ji,jj) * umask(ji,jj,jk-1)
673	ua(ji,jj,jk) = ua(ji,jj,jk) + ( zhpi(ji,jj,jk) + zuap) * umask(ji,jj,jk)
674
675	! hydrostatic pressure gradient along s-surfaces
676	! zhpi is masked for the first wet cell (contribution already done in the upper bloc)
677	zhpj(ji,jj,jk) = zhpj(ji,jj,jk) + zhpj(ji,jj,jk-1) &
678	& + zcoef0 / e2v(ji,jj) &
679	& * ( fse3w(ji ,jj+1,jk) * ( (rhd(ji,jj+1,jk ) + znad) &
680	& + (rhd(ji,jj+1,jk-1) + znad) ) * tmask(ji,jj+1,jk-1) &
681	& - fse3w(ji ,jj ,jk) * ( (rhd(ji,jj ,jk ) + znad) &
682	& + (rhd(ji,jj ,jk-1) + znad) ) * tmask(ji,jj ,jk-1) )
683	! s-coordinate pressure gradient correction
684	! corrective term, we mask this term for the first wet level beneath the ice shelf (contribution done in the upper bloc)
685	zvap = - zcoef0 * ( rhd (ji ,jj+1,jk) + rhd (ji,jj,jk) + 2._wp * znad ) &
686	& * ( fsde3w(ji ,jj+1,jk) - fsde3w(ji,jj,jk) ) / e2v(ji,jj) * vmask(ji,jj,jk-1)
687	! add to the general momentum trend
688	va(ji,jj,jk) = va(ji,jj,jk) + ( zhpj(ji,jj,jk) + zvap ) * vmask(ji,jj,jk)
689	END DO
690	END DO
691	END DO
692
693	!==================================================================================
694	!===== Compute bottom cell contribution (partial cell) ============================
695	!==================================================================================
696
697	DO jj = 2, jpjm1
698	DO ji = 2, jpim1
699	iku = mbku(ji,jj)
700	ikv = mbkv(ji,jj)
701
702	IF (iku .GT. 1) THEN
703	! remove old value (interior case)
704	zuap = -zcoef0 * ( rhd (ji+1,jj ,iku) + rhd (ji,jj,iku) + 2._wp * znad ) &
705	& * ( fsde3w(ji+1,jj ,iku) - fsde3w(ji,jj,iku) ) / e1u(ji,jj)
706	ua(ji,jj,iku) = ua(ji,jj,iku) - zhpi(ji,jj,iku) - zuap
707	! put new value
708	! -zpshpi to avoid double contribution of the partial step in the top layer
709	zuap = -zcoef0 * ( aru(ji,jj) + 2._wp * znad ) * gzu(ji,jj) / e1u(ji,jj)
710	zhpi(ji,jj,iku) = zhpi(ji,jj,iku-1) + zcoef0 / e1u(ji,jj) * ge3ru(ji,jj) - zpshpi(ji,jj)
711	ua(ji,jj,iku) = ua(ji,jj,iku) + zhpi(ji,jj,iku) + zuap
712	END IF
713	! v direction
714	IF (ikv .GT. 1) THEN
715	! remove old value (interior case)
716	zvap = -zcoef0 * ( rhd (ji ,jj+1,ikv) + rhd (ji,jj,ikv) + 2._wp * znad ) &
717	& * ( fsde3w(ji ,jj+1,ikv) - fsde3w(ji,jj,ikv) ) / e2v(ji,jj)
718	va(ji,jj,ikv) = va(ji,jj,ikv) - zhpj(ji,jj,ikv) - zvap
719	! put new value
720	! -zpshpj to avoid double contribution of the partial step in the top layer
721	zvap = -zcoef0 * ( arv(ji,jj) + 2._wp * znad ) * gzv(ji,jj) / e2v(ji,jj)
722	zhpj(ji,jj,ikv) = zhpj(ji,jj,ikv-1) + zcoef0 / e2v(ji,jj) * ge3rv(ji,jj) - zpshpj(ji,jj)
723	va(ji,jj,ikv) = va(ji,jj,ikv) + zhpj(ji,jj,ikv) + zvap
724	END IF
725	END DO
726	END DO
727
728	! set back to original density value into the ice shelf cell (maybe useless because it is masked)
729	rhd = zrhd
730	!
731	CALL wrk_dealloc( jpi,jpj,2, ztstop)
732	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zhpj, zrhd)
733	CALL wrk_dealloc( jpi,jpj, ze3w, zp, zrhdtop_isf, zrhdtop_oce, ziceload, zdept, zpshpi, zpshpj)
734	!
735	END SUBROUTINE hpg_isf
736
737
738	SUBROUTINE hpg_djc( kt )
739	!!---------------------------------------------------------------------
740	!! * ROUTINE hpg_djc *
741	!!
742	!! ** Method : Density Jacobian with Cubic polynomial scheme
743	!!
744	!! Reference: Shchepetkin and McWilliams, J. Geophys. Res., 108(C3), 3090, 2003
745	!!----------------------------------------------------------------------
746	INTEGER, INTENT(in) :: kt ! ocean time-step index
747	!!
748	INTEGER :: ji, jj, jk ! dummy loop indices
749	REAL(wp) :: zcoef0, zep, cffw ! temporary scalars
750	REAL(wp) :: z1_10, cffu, cffx ! " "
751	REAL(wp) :: z1_12, cffv, cffy ! " "
752	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zhpj
753	REAL(wp), POINTER, DIMENSION(:,:,:) :: dzx, dzy, dzz, dzu, dzv, dzw
754	REAL(wp), POINTER, DIMENSION(:,:,:) :: drhox, drhoy, drhoz, drhou, drhov, drhow
755	REAL(wp), POINTER, DIMENSION(:,:,:) :: rho_i, rho_j, rho_k
756	!!----------------------------------------------------------------------
757	!
758	CALL wrk_alloc( jpi, jpj, jpk, dzx , dzy , dzz , dzu , dzv , dzw )
759	CALL wrk_alloc( jpi, jpj, jpk, drhox, drhoy, drhoz, drhou, drhov, drhow )
760	CALL wrk_alloc( jpi, jpj, jpk, rho_i, rho_j, rho_k, zhpi, zhpj )
761	!
762
763	IF( kt == nit000 ) THEN
764	IF(lwp) WRITE(numout,*)
765	IF(lwp) WRITE(numout,*) 'dyn:hpg_djc : hydrostatic pressure gradient trend'
766	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, density Jacobian with cubic polynomial scheme'
767	ENDIF
768
769	! Local constant initialization
770	zcoef0 = - grav * 0.5_wp
771	z1_10 = 1._wp / 10._wp
772	z1_12 = 1._wp / 12._wp
773
774	!----------------------------------------------------------------------------------------
775	! compute and store in provisional arrays elementary vertical and horizontal differences
776	!----------------------------------------------------------------------------------------
777
778	!!bug gm Not a true bug, but... dzz=e3w for dzx, dzy verify what it is really
779
780	DO jk = 2, jpkm1
781	DO jj = 2, jpjm1
782	DO ji = fs_2, fs_jpim1 ! vector opt.
783	drhoz(ji,jj,jk) = rhd (ji ,jj ,jk) - rhd (ji,jj,jk-1)
784	dzz (ji,jj,jk) = fsde3w(ji ,jj ,jk) - fsde3w(ji,jj,jk-1)
785	drhox(ji,jj,jk) = rhd (ji+1,jj ,jk) - rhd (ji,jj,jk )
786	dzx (ji,jj,jk) = fsde3w(ji+1,jj ,jk) - fsde3w(ji,jj,jk )
787	drhoy(ji,jj,jk) = rhd (ji ,jj+1,jk) - rhd (ji,jj,jk )
788	dzy (ji,jj,jk) = fsde3w(ji ,jj+1,jk) - fsde3w(ji,jj,jk )
789	END DO
790	END DO
791	END DO
792
793	!-------------------------------------------------------------------------
794	! compute harmonic averages using eq. 5.18
795	!-------------------------------------------------------------------------
796	zep = 1.e-15
797
798	!!bug gm drhoz not defined at level 1 and used (jk-1 with jk=2)
799	!!bug gm idem for drhox, drhoy et ji=jpi and jj=jpj
800
801	DO jk = 2, jpkm1
802	DO jj = 2, jpjm1
803	DO ji = fs_2, fs_jpim1 ! vector opt.
804	cffw = 2._wp * drhoz(ji ,jj ,jk) * drhoz(ji,jj,jk-1)
805
806	cffu = 2._wp * drhox(ji+1,jj ,jk) * drhox(ji,jj,jk )
807	cffx = 2._wp * dzx (ji+1,jj ,jk) * dzx (ji,jj,jk )
808
809	cffv = 2._wp * drhoy(ji ,jj+1,jk) * drhoy(ji,jj,jk )
810	cffy = 2._wp * dzy (ji ,jj+1,jk) * dzy (ji,jj,jk )
811
812	IF( cffw > zep) THEN
813	drhow(ji,jj,jk) = 2._wp * drhoz(ji,jj,jk) * drhoz(ji,jj,jk-1) &
814	& / ( drhoz(ji,jj,jk) + drhoz(ji,jj,jk-1) )
815	ELSE
816	drhow(ji,jj,jk) = 0._wp
817	ENDIF
818
819	dzw(ji,jj,jk) = 2._wp * dzz(ji,jj,jk) * dzz(ji,jj,jk-1) &
820	& / ( dzz(ji,jj,jk) + dzz(ji,jj,jk-1) )
821
822	IF( cffu > zep ) THEN
823	drhou(ji,jj,jk) = 2._wp * drhox(ji+1,jj,jk) * drhox(ji,jj,jk) &
824	& / ( drhox(ji+1,jj,jk) + drhox(ji,jj,jk) )
825	ELSE
826	drhou(ji,jj,jk ) = 0._wp
827	ENDIF
828
829	IF( cffx > zep ) THEN
830	dzu(ji,jj,jk) = 2._wp * dzx(ji+1,jj,jk) * dzx(ji,jj,jk) &
831	& / ( dzx(ji+1,jj,jk) + dzx(ji,jj,jk) )
832	ELSE
833	dzu(ji,jj,jk) = 0._wp
834	ENDIF
835
836	IF( cffv > zep ) THEN
837	drhov(ji,jj,jk) = 2._wp * drhoy(ji,jj+1,jk) * drhoy(ji,jj,jk) &
838	& / ( drhoy(ji,jj+1,jk) + drhoy(ji,jj,jk) )
839	ELSE
840	drhov(ji,jj,jk) = 0._wp
841	ENDIF
842
843	IF( cffy > zep ) THEN
844	dzv(ji,jj,jk) = 2._wp * dzy(ji,jj+1,jk) * dzy(ji,jj,jk) &
845	& / ( dzy(ji,jj+1,jk) + dzy(ji,jj,jk) )
846	ELSE
847	dzv(ji,jj,jk) = 0._wp
848	ENDIF
849
850	END DO
851	END DO
852	END DO
853
854	!----------------------------------------------------------------------------------
855	! apply boundary conditions at top and bottom using 5.36-5.37
856	!----------------------------------------------------------------------------------
857	drhow(:,:, 1 ) = 1.5_wp * ( drhoz(:,:, 2 ) - drhoz(:,:, 1 ) ) - 0.5_wp * drhow(:,:, 2 )
858	drhou(:,:, 1 ) = 1.5_wp * ( drhox(:,:, 2 ) - drhox(:,:, 1 ) ) - 0.5_wp * drhou(:,:, 2 )
859	drhov(:,:, 1 ) = 1.5_wp * ( drhoy(:,:, 2 ) - drhoy(:,:, 1 ) ) - 0.5_wp * drhov(:,:, 2 )
860
861	drhow(:,:,jpk) = 1.5_wp * ( drhoz(:,:,jpk) - drhoz(:,:,jpkm1) ) - 0.5_wp * drhow(:,:,jpkm1)
862	drhou(:,:,jpk) = 1.5_wp * ( drhox(:,:,jpk) - drhox(:,:,jpkm1) ) - 0.5_wp * drhou(:,:,jpkm1)
863	drhov(:,:,jpk) = 1.5_wp * ( drhoy(:,:,jpk) - drhoy(:,:,jpkm1) ) - 0.5_wp * drhov(:,:,jpkm1)
864
865
866	!--------------------------------------------------------------
867	! Upper half of top-most grid box, compute and store
868	!-------------------------------------------------------------
869
870	!!bug gm : e3w-de3w = 0.5*e3w .... and de3w(2)-de3w(1)=e3w(2) .... to be verified
871	! true if de3w is really defined as the sum of the e3w scale factors as, it seems to me, it should be
872
873	DO jj = 2, jpjm1
874	DO ji = fs_2, fs_jpim1 ! vector opt.
875	rho_k(ji,jj,1) = -grav * ( fse3w(ji,jj,1) - fsde3w(ji,jj,1) ) &
876	& * ( rhd(ji,jj,1) &
877	& + 0.5_wp * ( rhd(ji,jj,2) - rhd(ji,jj,1) ) &
878	& * ( fse3w (ji,jj,1) - fsde3w(ji,jj,1) ) &
879	& / ( fsde3w(ji,jj,2) - fsde3w(ji,jj,1) ) )
880	END DO
881	END DO
882
883	!!bug gm : here also, simplification is possible
884	!!bug gm : optimisation: 1/10 and 1/12 the division should be done before the loop
885
886	DO jk = 2, jpkm1
887	DO jj = 2, jpjm1
888	DO ji = fs_2, fs_jpim1 ! vector opt.
889
890	rho_k(ji,jj,jk) = zcoef0 * ( rhd (ji,jj,jk) + rhd (ji,jj,jk-1) ) &
891	& * ( fsde3w(ji,jj,jk) - fsde3w(ji,jj,jk-1) ) &
892	& - grav * z1_10 * ( &
893	& ( drhow (ji,jj,jk) - drhow (ji,jj,jk-1) ) &
894	& * ( fsde3w(ji,jj,jk) - fsde3w(ji,jj,jk-1) - z1_12 * ( dzw (ji,jj,jk) + dzw (ji,jj,jk-1) ) ) &
895	& - ( dzw (ji,jj,jk) - dzw (ji,jj,jk-1) ) &
896	& * ( rhd (ji,jj,jk) - rhd (ji,jj,jk-1) - z1_12 * ( drhow(ji,jj,jk) + drhow(ji,jj,jk-1) ) ) &
897	& )
898
899	rho_i(ji,jj,jk) = zcoef0 * ( rhd (ji+1,jj,jk) + rhd (ji,jj,jk) ) &
900	& * ( fsde3w(ji+1,jj,jk) - fsde3w(ji,jj,jk) ) &
901	& - grav* z1_10 * ( &
902	& ( drhou (ji+1,jj,jk) - drhou (ji,jj,jk) ) &
903	& * ( fsde3w(ji+1,jj,jk) - fsde3w(ji,jj,jk) - z1_12 * ( dzu (ji+1,jj,jk) + dzu (ji,jj,jk) ) ) &
904	& - ( dzu (ji+1,jj,jk) - dzu (ji,jj,jk) ) &
905	& * ( rhd (ji+1,jj,jk) - rhd (ji,jj,jk) - z1_12 * ( drhou(ji+1,jj,jk) + drhou(ji,jj,jk) ) ) &
906	& )
907
908	rho_j(ji,jj,jk) = zcoef0 * ( rhd (ji,jj+1,jk) + rhd (ji,jj,jk) ) &
909	& * ( fsde3w(ji,jj+1,jk) - fsde3w(ji,jj,jk) ) &
910	& - grav* z1_10 * ( &
911	& ( drhov (ji,jj+1,jk) - drhov (ji,jj,jk) ) &
912	& * ( fsde3w(ji,jj+1,jk) - fsde3w(ji,jj,jk) - z1_12 * ( dzv (ji,jj+1,jk) + dzv (ji,jj,jk) ) ) &
913	& - ( dzv (ji,jj+1,jk) - dzv (ji,jj,jk) ) &
914	& * ( rhd (ji,jj+1,jk) - rhd (ji,jj,jk) - z1_12 * ( drhov(ji,jj+1,jk) + drhov(ji,jj,jk) ) ) &
915	& )
916
917	END DO
918	END DO
919	END DO
920	CALL lbc_lnk(rho_k,'W',1.)
921	CALL lbc_lnk(rho_i,'U',1.)
922	CALL lbc_lnk(rho_j,'V',1.)
923
924
925	! ---------------
926	! Surface value
927	! ---------------
928	DO jj = 2, jpjm1
929	DO ji = fs_2, fs_jpim1 ! vector opt.
930	zhpi(ji,jj,1) = ( rho_k(ji+1,jj ,1) - rho_k(ji,jj,1) - rho_i(ji,jj,1) ) / e1u(ji,jj)
931	zhpj(ji,jj,1) = ( rho_k(ji ,jj+1,1) - rho_k(ji,jj,1) - rho_j(ji,jj,1) ) / e2v(ji,jj)
932	! add to the general momentum trend
933	ua(ji,jj,1) = ua(ji,jj,1) + zhpi(ji,jj,1)
934	va(ji,jj,1) = va(ji,jj,1) + zhpj(ji,jj,1)
935	END DO
936	END DO
937
938	! ----------------
939	! interior value (2=<jk=<jpkm1)
940	! ----------------
941	DO jk = 2, jpkm1
942	DO jj = 2, jpjm1
943	DO ji = fs_2, fs_jpim1 ! vector opt.
944	! hydrostatic pressure gradient along s-surfaces
945	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) &
946	& + ( ( rho_k(ji+1,jj,jk) - rho_k(ji,jj,jk ) ) &
947	& - ( rho_i(ji ,jj,jk) - rho_i(ji,jj,jk-1) ) ) / e1u(ji,jj)
948	zhpj(ji,jj,jk) = zhpj(ji,jj,jk-1) &
949	& + ( ( rho_k(ji,jj+1,jk) - rho_k(ji,jj,jk ) ) &
950	& -( rho_j(ji,jj ,jk) - rho_j(ji,jj,jk-1) ) ) / e2v(ji,jj)
951	! add to the general momentum trend
952	ua(ji,jj,jk) = ua(ji,jj,jk) + zhpi(ji,jj,jk)
953	va(ji,jj,jk) = va(ji,jj,jk) + zhpj(ji,jj,jk)
954	END DO
955	END DO
956	END DO
957	!
958	CALL wrk_dealloc( jpi, jpj, jpk, dzx , dzy , dzz , dzu , dzv , dzw )
959	CALL wrk_dealloc( jpi, jpj, jpk, drhox, drhoy, drhoz, drhou, drhov, drhow )
960	CALL wrk_dealloc( jpi, jpj, jpk, rho_i, rho_j, rho_k, zhpi, zhpj )
961	!
962	END SUBROUTINE hpg_djc
963
964
965	SUBROUTINE hpg_prj( kt )
966	!!---------------------------------------------------------------------
967	!! * ROUTINE hpg_prj *
968	!!
969	!! ** Method : s-coordinate case.
970	!! A Pressure-Jacobian horizontal pressure gradient method
971	!! based on the constrained cubic-spline interpolation for
972	!! all vertical coordinate systems
973	!!
974	!! ** Action : - Update (ua,va) with the now hydrastatic pressure trend
975	!!----------------------------------------------------------------------
976	INTEGER, PARAMETER :: polynomial_type = 1 ! 1: cubic spline, 2: linear
977	INTEGER, INTENT(in) :: kt ! ocean time-step index
978	!!
979	INTEGER :: ji, jj, jk, jkk ! dummy loop indices
980	REAL(wp) :: zcoef0, znad ! temporary scalars
981	!!
982	!! The local variables for the correction term
983	INTEGER :: jk1, jis, jid, jjs, jjd
984	REAL(wp) :: zuijk, zvijk, zpwes, zpwed, zpnss, zpnsd, zdeps
985	REAL(wp) :: zrhdt1
986	REAL(wp) :: zdpdx1, zdpdx2, zdpdy1, zdpdy2
987	REAL(wp), POINTER, DIMENSION(:,:,:) :: zdept, zrhh
988	REAL(wp), POINTER, DIMENSION(:,:,:) :: zhpi, zu, zv, fsp, xsp, asp, bsp, csp, dsp
989	REAL(wp), POINTER, DIMENSION(:,:) :: zsshu_n, zsshv_n
990	!!----------------------------------------------------------------------
991	!
992	CALL wrk_alloc( jpi,jpj,jpk, zhpi, zu, zv, fsp, xsp, asp, bsp, csp, dsp )
993	CALL wrk_alloc( jpi,jpj,jpk, zdept, zrhh )
994	CALL wrk_alloc( jpi,jpj, zsshu_n, zsshv_n )
995	!
996	IF( kt == nit000 ) THEN
997	IF(lwp) WRITE(numout,*)
998	IF(lwp) WRITE(numout,*) 'dyn:hpg_prj : hydrostatic pressure gradient trend'
999	IF(lwp) WRITE(numout,*) '~~~~~~~~~~~ s-coordinate case, cubic spline pressure Jacobian'
1000	ENDIF
1001
1002	!!----------------------------------------------------------------------
1003	! Local constant initialization
1004	zcoef0 = - grav
1005	znad = 0.0_wp
1006	IF( lk_vvl ) znad = 1._wp
1007
1008	! Clean 3-D work arrays
1009	zhpi(:,:,:) = 0._wp
1010	zrhh(:,:,:) = rhd(:,:,:)
1011
1012	! Preparing vertical density profile "zrhh(:,:,:)" for hybrid-sco coordinate
1013	DO jj = 1, jpj
1014	DO ji = 1, jpi
1015	jk = mbathy(ji,jj)
1016	IF( jk <= 0 ) THEN; zrhh(ji,jj,:) = 0._wp
1017	ELSE IF(jk == 1) THEN; zrhh(ji,jj, jk+1:jpk) = rhd(ji,jj,jk)
1018	ELSE IF(jk < jpkm1) THEN
1019	DO jkk = jk+1, jpk
1020	zrhh(ji,jj,jkk) = interp1(fsde3w(ji,jj,jkk), fsde3w(ji,jj,jkk-1), &
1021	fsde3w(ji,jj,jkk-2), rhd(ji,jj,jkk-1), rhd(ji,jj,jkk-2))
1022	END DO
1023	ENDIF
1024	END DO
1025	END DO
1026
1027	! Transfer the depth of "T(:,:,:)" to vertical coordinate "zdept(:,:,:)"
1028	DO jj = 1, jpj
1029	DO ji = 1, jpi
1030	zdept(ji,jj,1) = 0.5_wp * fse3w(ji,jj,1) - sshn(ji,jj) * znad
1031	END DO
1032	END DO
1033
1034	DO jk = 2, jpk
1035	DO jj = 1, jpj
1036	DO ji = 1, jpi
1037	zdept(ji,jj,jk) = zdept(ji,jj,jk-1) + fse3w(ji,jj,jk)
1038	END DO
1039	END DO
1040	END DO
1041
1042	fsp(:,:,:) = zrhh (:,:,:)
1043	xsp(:,:,:) = zdept(:,:,:)
1044
1045	! Construct the vertical density profile with the
1046	! constrained cubic spline interpolation
1047	! rho(z) = asp + bspz + cspz^2 + dsp*z^3
1048	CALL cspline(fsp,xsp,asp,bsp,csp,dsp,polynomial_type)
1049
1050	! Integrate the hydrostatic pressure "zhpi(:,:,:)" at "T(ji,jj,1)"
1051	DO jj = 2, jpj
1052	DO ji = 2, jpi
1053	zrhdt1 = zrhh(ji,jj,1) - interp3(zdept(ji,jj,1),asp(ji,jj,1), &
1054	bsp(ji,jj,1), csp(ji,jj,1), &
1055	dsp(ji,jj,1) ) * 0.25_wp * fse3w(ji,jj,1)
1056
1057	! assuming linear profile across the top half surface layer
1058	zhpi(ji,jj,1) = 0.5_wp * fse3w(ji,jj,1) * zrhdt1
1059	END DO
1060	END DO
1061
1062	! Calculate the pressure "zhpi(:,:,:)" at "T(ji,jj,2:jpkm1)"
1063	DO jk = 2, jpkm1
1064	DO jj = 2, jpj
1065	DO ji = 2, jpi
1066	zhpi(ji,jj,jk) = zhpi(ji,jj,jk-1) + &
1067	integ_spline(zdept(ji,jj,jk-1), zdept(ji,jj,jk),&
1068	asp(ji,jj,jk-1), bsp(ji,jj,jk-1), &
1069	csp(ji,jj,jk-1), dsp(ji,jj,jk-1))
1070	END DO
1071	END DO
1072	END DO
1073
1074	! Z coordinate of U(ji,jj,1:jpkm1) and V(ji,jj,1:jpkm1)
1075
1076	! Prepare zsshu_n and zsshv_n
1077	DO jj = 2, jpjm1
1078	DO ji = 2, jpim1
1079	zsshu_n(ji,jj) = (e12u(ji,jj) * sshn(ji,jj) + e12u(ji+1, jj) * sshn(ji+1,jj)) * &
1080	& r1_e12u(ji,jj) * umask(ji,jj,1) * 0.5_wp
1081	zsshv_n(ji,jj) = (e12v(ji,jj) * sshn(ji,jj) + e12v(ji+1, jj) * sshn(ji,jj+1)) * &
1082	& r1_e12v(ji,jj) * vmask(ji,jj,1) * 0.5_wp
1083	END DO
1084	END DO
1085
1086	DO jj = 2, jpjm1
1087	DO ji = 2, jpim1
1088	zu(ji,jj,1) = - ( fse3u(ji,jj,1) - zsshu_n(ji,jj) * znad)
1089	zv(ji,jj,1) = - ( fse3v(ji,jj,1) - zsshv_n(ji,jj) * znad)
1090	END DO
1091	END DO
1092
1093	DO jk = 2, jpkm1
1094	DO jj = 2, jpjm1
1095	DO ji = 2, jpim1
1096	zu(ji,jj,jk) = zu(ji,jj,jk-1)- fse3u(ji,jj,jk)
1097	zv(ji,jj,jk) = zv(ji,jj,jk-1)- fse3v(ji,jj,jk)
1098	END DO
1099	END DO
1100	END DO
1101
1102	DO jk = 1, jpkm1
1103	DO jj = 2, jpjm1
1104	DO ji = 2, jpim1
1105	zu(ji,jj,jk) = zu(ji,jj,jk) + 0.5_wp * fse3u(ji,jj,jk)
1106	zv(ji,jj,jk) = zv(ji,jj,jk) + 0.5_wp * fse3v(ji,jj,jk)
1107	END DO
1108	END DO
1109	END DO
1110
1111	DO jk = 1, jpkm1
1112	DO jj = 2, jpjm1
1113	DO ji = 2, jpim1
1114	zu(ji,jj,jk) = min(zu(ji,jj,jk), max(-zdept(ji,jj,jk), -zdept(ji+1,jj,jk)))
1115	zu(ji,jj,jk) = max(zu(ji,jj,jk), min(-zdept(ji,jj,jk), -zdept(ji+1,jj,jk)))
1116	zv(ji,jj,jk) = min(zv(ji,jj,jk), max(-zdept(ji,jj,jk), -zdept(ji,jj+1,jk)))
1117	zv(ji,jj,jk) = max(zv(ji,jj,jk), min(-zdept(ji,jj,jk), -zdept(ji,jj+1,jk)))
1118	END DO
1119	END DO
1120	END DO
1121
1122
1123	DO jk = 1, jpkm1
1124	DO jj = 2, jpjm1
1125	DO ji = 2, jpim1
1126	zpwes = 0._wp; zpwed = 0._wp
1127	zpnss = 0._wp; zpnsd = 0._wp
1128	zuijk = zu(ji,jj,jk)
1129	zvijk = zv(ji,jj,jk)
1130
1131	!!!!! for u equation
1132	IF( jk <= mbku(ji,jj) ) THEN
1133	IF( -zdept(ji+1,jj,jk) >= -zdept(ji,jj,jk) ) THEN
1134	jis = ji + 1; jid = ji
1135	ELSE
1136	jis = ji; jid = ji +1
1137	ENDIF
1138
1139	! integrate the pressure on the shallow side
1140	jk1 = jk
1141	DO WHILE ( -zdept(jis,jj,jk1) > zuijk )
1142	IF( jk1 == mbku(ji,jj) ) THEN
1143	zuijk = -zdept(jis,jj,jk1)
1144	EXIT
1145	ENDIF
1146	zdeps = MIN(zdept(jis,jj,jk1+1), -zuijk)
1147	zpwes = zpwes + &
1148	integ_spline(zdept(jis,jj,jk1), zdeps, &
1149	asp(jis,jj,jk1), bsp(jis,jj,jk1), &
1150	csp(jis,jj,jk1), dsp(jis,jj,jk1))
1151	jk1 = jk1 + 1
1152	END DO
1153
1154	! integrate the pressure on the deep side
1155	jk1 = jk
1156	DO WHILE ( -zdept(jid,jj,jk1) < zuijk )
1157	IF( jk1 == 1 ) THEN
1158	zdeps = zdept(jid,jj,1) + MIN(zuijk, sshn(jid,jj)*znad)
1159	zrhdt1 = zrhh(jid,jj,1) - interp3(zdept(jid,jj,1), asp(jid,jj,1), &
1160	bsp(jid,jj,1), csp(jid,jj,1), &
1161	dsp(jid,jj,1)) * zdeps
1162	zpwed = zpwed + 0.5_wp * (zrhh(jid,jj,1) + zrhdt1) * zdeps
1163	EXIT
1164	ENDIF
1165	zdeps = MAX(zdept(jid,jj,jk1-1), -zuijk)
1166	zpwed = zpwed + &
1167	integ_spline(zdeps, zdept(jid,jj,jk1), &
1168	asp(jid,jj,jk1-1), bsp(jid,jj,jk1-1), &
1169	csp(jid,jj,jk1-1), dsp(jid,jj,jk1-1) )
1170	jk1 = jk1 - 1
1171	END DO
1172
1173	! update the momentum trends in u direction
1174
1175	zdpdx1 = zcoef0 / e1u(ji,jj) * (zhpi(ji+1,jj,jk) - zhpi(ji,jj,jk))
1176	IF( lk_vvl ) THEN
1177	zdpdx2 = zcoef0 / e1u(ji,jj) * &
1178	( REAL(jis-jid, wp) * (zpwes + zpwed) + (sshn(ji+1,jj)-sshn(ji,jj)) )
1179	ELSE
1180	zdpdx2 = zcoef0 / e1u(ji,jj) * REAL(jis-jid, wp) * (zpwes + zpwed)
1181	ENDIF
1182
1183	ua(ji,jj,jk) = ua(ji,jj,jk) + (zdpdx1 + zdpdx2) * &
1184	& umask(ji,jj,jk) * tmask(ji,jj,jk) * tmask(ji+1,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, sshn(ji,jjd)*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 / e2v(ji,jj) * (zhpi(ji,jj+1,jk) - zhpi(ji,jj,jk))
1233	IF( lk_vvl ) THEN
1234	zdpdy2 = zcoef0 / e2v(ji,jj) * &
1235	( REAL(jjs-jjd, wp) * (zpnss + zpnsd) + (sshn(ji,jj+1)-sshn(ji,jj)) )
1236	ELSE
1237	zdpdy2 = zcoef0 / e2v(ji,jj) * REAL(jjs-jjd, wp) * (zpnss + zpnsd )
1238	ENDIF
1239
1240	va(ji,jj,jk) = va(ji,jj,jk) + (zdpdy1 + zdpdy2)*&
1241	& vmask(ji,jj,jk)tmask(ji,jj,jk)tmask(ji,jj+1,jk)
1242	ENDIF
1243
1244
1245	END DO
1246	END DO
1247	END DO
1248	!
1249	CALL wrk_dealloc( jpi,jpj,jpk, zhpi, zu, zv, fsp, xsp, asp, bsp, csp, dsp )
1250	CALL wrk_dealloc( jpi,jpj,jpk, zdept, zrhh )
1251	CALL wrk_dealloc( jpi,jpj, zsshu_n, zsshv_n )
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	IMPLICIT NONE
1267	REAL(wp), DIMENSION(:,:,:), INTENT(in) :: fsp, xsp ! value and coordinate
1268	REAL(wp), DIMENSION(:,:,:), INTENT(out) :: asp, bsp, csp, dsp ! coefficients of
1269	! the interpoated function
1270	INTEGER, INTENT(in) :: polynomial_type ! 1: cubic spline
1271	! 2: Linear
1272	!
1273	INTEGER :: ji, jj, jk ! dummy loop indices
1274	INTEGER :: jpi, jpj, jpkm1
1275	REAL(wp) :: zdf1, zdf2, zddf1, zddf2, ztmp1, ztmp2, zdxtmp
1276	REAL(wp) :: zdxtmp1, zdxtmp2, zalpha
1277	REAL(wp) :: zdf(size(fsp,3))
1278	!!----------------------------------------------------------------------
1279
1280	jpi = size(fsp,1)
1281	jpj = size(fsp,2)
1282	jpkm1 = size(fsp,3) - 1
1283
1284
1285	IF (polynomial_type == 1) THEN ! Constrained Cubic Spline
1286	DO ji = 1, jpi
1287	DO jj = 1, jpj
1288	!!Fritsch&Butland's method, 1984 (preferred, but more computation)
1289	! DO jk = 2, jpkm1-1
1290	! zdxtmp1 = xsp(ji,jj,jk) - xsp(ji,jj,jk-1)
1291	! zdxtmp2 = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1292	! zdf1 = ( fsp(ji,jj,jk) - fsp(ji,jj,jk-1) ) / zdxtmp1
1293	! zdf2 = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp2
1294	!
1295	! zalpha = ( zdxtmp1 + 2._wp * zdxtmp2 ) / ( zdxtmp1 + zdxtmp2 ) / 3._wp
1296	!
1297	! IF(zdf1 * zdf2 <= 0._wp) THEN
1298	! zdf(jk) = 0._wp
1299	! ELSE
1300	! zdf(jk) = zdf1 * zdf2 / ( ( 1._wp - zalpha ) * zdf1 + zalpha * zdf2 )
1301	! ENDIF
1302	! END DO
1303
1304	!!Simply geometric average
1305	DO jk = 2, jpkm1-1
1306	zdf1 = (fsp(ji,jj,jk) - fsp(ji,jj,jk-1)) / (xsp(ji,jj,jk) - xsp(ji,jj,jk-1))
1307	zdf2 = (fsp(ji,jj,jk+1) - fsp(ji,jj,jk)) / (xsp(ji,jj,jk+1) - xsp(ji,jj,jk))
1308
1309	IF(zdf1 * zdf2 <= 0._wp) THEN
1310	zdf(jk) = 0._wp
1311	ELSE
1312	zdf(jk) = 2._wp * zdf1 * zdf2 / (zdf1 + zdf2)
1313	ENDIF
1314	END DO
1315
1316	zdf(1) = 1.5_wp * ( fsp(ji,jj,2) - fsp(ji,jj,1) ) / &
1317	& ( xsp(ji,jj,2) - xsp(ji,jj,1) ) - 0.5_wp * zdf(2)
1318	zdf(jpkm1) = 1.5_wp * ( fsp(ji,jj,jpkm1) - fsp(ji,jj,jpkm1-1) ) / &
1319	& ( xsp(ji,jj,jpkm1) - xsp(ji,jj,jpkm1-1) ) - &
1320	& 0.5_wp * zdf(jpkm1 - 1)
1321
1322	DO jk = 1, jpkm1 - 1
1323	zdxtmp = xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1324	ztmp1 = (zdf(jk+1) + 2._wp * zdf(jk)) / zdxtmp
1325	ztmp2 = 6._wp * (fsp(ji,jj,jk+1) - fsp(ji,jj,jk)) / zdxtmp / zdxtmp
1326	zddf1 = -2._wp * ztmp1 + ztmp2
1327	ztmp1 = (2._wp * zdf(jk+1) + zdf(jk)) / zdxtmp
1328	zddf2 = 2._wp * ztmp1 - ztmp2
1329
1330	dsp(ji,jj,jk) = (zddf2 - zddf1) / 6._wp / zdxtmp
1331	csp(ji,jj,jk) = ( xsp(ji,jj,jk+1) * zddf1 - xsp(ji,jj,jk)*zddf2 ) / 2._wp / zdxtmp
1332	bsp(ji,jj,jk) = ( fsp(ji,jj,jk+1) - fsp(ji,jj,jk) ) / zdxtmp - &
1333	& csp(ji,jj,jk) * ( xsp(ji,jj,jk+1) + xsp(ji,jj,jk) ) - &
1334	& dsp(ji,jj,jk) * ((xsp(ji,jj,jk+1) + xsp(ji,jj,jk))**2 - &
1335	& xsp(ji,jj,jk+1) * xsp(ji,jj,jk))
1336	asp(ji,jj,jk) = fsp(ji,jj,jk) - xsp(ji,jj,jk) * (bsp(ji,jj,jk) + &
1337	& (xsp(ji,jj,jk) * (csp(ji,jj,jk) + &
1338	& dsp(ji,jj,jk) * xsp(ji,jj,jk))))
1339	END DO
1340	END DO
1341	END DO
1342
1343	ELSE IF (polynomial_type == 2) THEN ! Linear
1344	DO ji = 1, jpi
1345	DO jj = 1, jpj
1346	DO jk = 1, jpkm1-1
1347	zdxtmp =xsp(ji,jj,jk+1) - xsp(ji,jj,jk)
1348	ztmp1 = fsp(ji,jj,jk+1) - fsp(ji,jj,jk)
1349
1350	dsp(ji,jj,jk) = 0._wp
1351	csp(ji,jj,jk) = 0._wp
1352	bsp(ji,jj,jk) = ztmp1 / zdxtmp
1353	asp(ji,jj,jk) = fsp(ji,jj,jk) - bsp(ji,jj,jk) * xsp(ji,jj,jk)
1354	END DO
1355	END DO
1356	END DO
1357
1358	ELSE
1359	CALL ctl_stop( 'invalid polynomial type in cspline' )
1360	ENDIF
1361
1362	END SUBROUTINE cspline
1363
1364
1365	FUNCTION interp1(x, xl, xr, fl, fr) RESULT(f)
1366	!!----------------------------------------------------------------------
1367	!! * ROUTINE interp1 *
1368	!!
1369	!! ** Purpose : 1-d linear interpolation
1370	!!
1371	!! ** Method : interpolation is straight forward
1372	!! extrapolation is also permitted (no value limit)
1373	!!----------------------------------------------------------------------
1374	IMPLICIT NONE
1375	REAL(wp), INTENT(in) :: x, xl, xr, fl, fr
1376	REAL(wp) :: f ! result of the interpolation (extrapolation)
1377	REAL(wp) :: zdeltx
1378	!!----------------------------------------------------------------------
1379
1380	zdeltx = xr - xl
1381	IF(abs(zdeltx) <= 10._wp * EPSILON(x)) THEN
1382	f = 0.5_wp * (fl + fr)
1383	ELSE
1384	f = ( (x - xl ) * fr - ( x - xr ) * fl ) / zdeltx
1385	ENDIF
1386
1387	END FUNCTION interp1
1388
1389
1390	FUNCTION interp2(x, a, b, c, d) RESULT(f)
1391	!!----------------------------------------------------------------------
1392	!! * ROUTINE interp1 *
1393	!!
1394	!! ** Purpose : 1-d constrained cubic spline interpolation
1395	!!
1396	!! ** Method : cubic spline interpolation
1397	!!
1398	!!----------------------------------------------------------------------
1399	IMPLICIT NONE
1400	REAL(wp), INTENT(in) :: x, a, b, c, d
1401	REAL(wp) :: f ! value from the interpolation
1402	!!----------------------------------------------------------------------
1403
1404	f = a + x* ( b + x * ( c + d * x ) )
1405
1406	END FUNCTION interp2
1407
1408
1409	FUNCTION interp3(x, a, b, c, d) RESULT(f)
1410	!!----------------------------------------------------------------------
1411	!! * ROUTINE interp1 *
1412	!!
1413	!! ** Purpose : Calculate the first order of deriavtive of
1414	!! a cubic spline function y=a+bx+cx^2+d*x^3
1415	!!
1416	!! ** Method : f=dy/dx=b+2cx+3dx^2
1417	!!
1418	!!----------------------------------------------------------------------
1419	IMPLICIT NONE
1420	REAL(wp), INTENT(in) :: x, a, b, c, d
1421	REAL(wp) :: f ! value from the interpolation
1422	!!----------------------------------------------------------------------
1423
1424	f = b + x * ( 2._wp * c + 3._wp * d * x)
1425
1426	END FUNCTION interp3
1427
1428
1429	FUNCTION integ_spline(xl, xr, a, b, c, d) RESULT(f)
1430	!!----------------------------------------------------------------------
1431	!! * ROUTINE interp1 *
1432	!!
1433	!! ** Purpose : 1-d constrained cubic spline integration
1434	!!
1435	!! ** Method : integrate polynomial a+bx+cx^2+dx^3 from xl to xr
1436	!!
1437	!!----------------------------------------------------------------------
1438	IMPLICIT NONE
1439	REAL(wp), INTENT(in) :: xl, xr, a, b, c, d
1440	REAL(wp) :: za1, za2, za3
1441	REAL(wp) :: f ! integration result
1442	!!----------------------------------------------------------------------
1443
1444	za1 = 0.5_wp * b
1445	za2 = c / 3.0_wp
1446	za3 = 0.25_wp * d
1447
1448	f = xr * ( a + xr * ( za1 + xr * ( za2 + za3 * xr ) ) ) - &
1449	& xl * ( a + xl * ( za1 + xl * ( za2 + za3 * xl ) ) )
1450
1451	END FUNCTION integ_spline
1452
1453	!!======================================================================
1454	END MODULE dynhpg
1455

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