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Last change on this file since 4651 was 4509, checked in by raasch, 5 years ago
files re-formatted to follow the PALM coding standard
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1	!> @file basic_constants_and_equations_mod.f90
2	!--------------------------------------------------------------------------------------------------!
3	! This file is part of the PALM model system.
4	!
5	! PALM is free software: you can redistribute it and/or modify it under the terms of the GNU General
6	! Public License as published by the Free Software Foundation, either version 3 of the License, or
7	! (at your option) any later version.
8	!
9	! PALM is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the
10	! implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General
11	! Public License for more details.
12	!
13	! You should have received a copy of the GNU General Public License along with PALM. If not, see
14	! <http://www.gnu.org/licenses/>.
15	!
16	! Copyright 1997-2020 Leibniz Universitaet Hannover
17	!--------------------------------------------------------------------------------------------------!
18	!
19	! Current revisions:
20	! -----------------
21	!
22	!
23	! Former revisions:
24	! -----------------
25	! $Id: basic_constants_and_equations_mod.f90 4509 2020-04-26 15:57:55Z raasch $
26	! file re-formatted to follow the PALM coding standard
27	!
28	! 4502 2020-04-17 16:14:16Z schwenkel
29	! Implementation of ice microphysics
30	!
31	! 4400 2020-02-10 20:32:41Z suehring
32	! Move routine to transform coordinates from netcdf_interface_mod to
33	! basic_constants_and_equations_mod
34	!
35	! 4360 2020-01-07 11:25:50Z suehring
36	! Corrected "Former revisions" section
37	!
38	! 4088 2019-07-11 13:57:56Z Giersch
39	! Comment of barometric formula improved, function for ideal gas law revised
40	!
41	! 4084 2019-07-10 17:09:11Z knoop
42	! Changed precomputed fractions to be variable based
43	!
44	! 4055 2019-06-27 09:47:29Z suehring
45	! Added rgas_univ (universal gas constant) (E.C. Chan)
46	!
47	!
48	! 3655 2019-01-07 16:51:22Z knoop
49	! OpenACC port for SPEC
50	! 3361 2018-10-16 20:39:37Z knoop
51	! New module (introduced with modularization of bulk cloud physics model)
52	!
53	!
54	!
55	!
56	! Description:
57	! ------------
58	!> This module contains all basic (physical) constants and functions for the calculation of
59	!> diagnostic quantities.
60	!- -----------------------------------------------------------------------------!
61	MODULE basic_constants_and_equations_mod
62
63
64	USE kinds
65
66	IMPLICIT NONE
67
68
69	REAL(wp), PARAMETER :: c_p = 1005.0_wp !< heat capacity of dry air (J kg-1 K-1)
70	REAL(wp), PARAMETER :: degc_to_k = 273.15_wp !< temperature (in K) of 0 deg C (K)
71	REAL(wp), PARAMETER :: g = 9.81_wp !< gravitational acceleration (m s-2)
72	REAL(wp), PARAMETER :: kappa = 0.4_wp !< von Karman constant
73	REAL(wp), PARAMETER :: l_m = 0.33E+06_wp !< latent heat of water melting (J kg-1)
74	REAL(wp), PARAMETER :: l_v = 2.5E+06_wp !< latent heat of water vaporization (J kg-1)
75	REAL(wp), PARAMETER :: l_s = l_m + l_v !< latent heat of water sublimation (J kg-1)
76	REAL(wp), PARAMETER :: molecular_weight_of_nacl = 0.05844_wp !< mol. m. NaCl (kg mol-1)
77	REAL(wp), PARAMETER :: molecular_weight_of_c3h4o4 = 0.10406_wp !< mol. m. malonic acid (kg mol-1)
78	REAL(wp), PARAMETER :: molecular_weight_of_nh4no3 = 0.08004_wp !< mol. m. ammonium sulfate (kg mol-1)
79	REAL(wp), PARAMETER :: molecular_weight_of_water = 0.01801528_wp !< mol. m. H2O (kg mol-1)
80	REAL(wp), PARAMETER :: pi = 3.141592654_wp !< PI
81	!$ACC DECLARE COPYIN(pi)
82	REAL(wp), PARAMETER :: rgas_univ = 8.31446261815324_wp !< universal gas constant (J K-1 mol-1)
83	REAL(wp), PARAMETER :: rho_l = 1.0E3_wp !< density of water (kg m-3)
84	REAL(wp), PARAMETER :: rho_nacl = 2165.0_wp !< density of NaCl (kg m-3)
85	REAL(wp), PARAMETER :: rho_c3h4o4 = 1600.0_wp !< density of malonic acid (kg m-3)
86	REAL(wp), PARAMETER :: rho_nh4no3 = 1720.0_wp !< density of ammonium sulfate (kg m-3)
87	REAL(wp), PARAMETER :: r_d = 287.0_wp !< sp. gas const. dry air (J kg-1 K-1)
88	REAL(wp), PARAMETER :: r_v = 461.51_wp !< sp. gas const. water vapor (J kg-1 K-1)
89	REAL(wp), PARAMETER :: sigma_sb = 5.67037E-08_wp !< Stefan-Boltzmann constant
90	REAL(wp), PARAMETER :: solar_constant = 1368.0_wp !< solar constant at top of atmosphere
91	REAL(wp), PARAMETER :: vanthoff_nacl = 2.0_wp !< van't Hoff factor for NaCl
92	REAL(wp), PARAMETER :: vanthoff_c3h4o4 = 1.37_wp !< van't Hoff factor for malonic acid
93	REAL(wp), PARAMETER :: vanthoff_nh4no3 = 2.31_wp !< van't Hoff factor for ammonium sulfate
94
95	REAL(wp), PARAMETER :: p_0 = 100000.0_wp !< standard pressure reference state
96
97	REAL(wp), PARAMETER :: cp_d_rd = c_p / r_d !< precomputed c_p / r_d
98	REAL(wp), PARAMETER :: g_d_cp = g / c_p !< precomputed g / c_p
99	REAL(wp), PARAMETER :: lv_d_cp = l_v / c_p !< precomputed l_v / c_p
100	REAL(wp), PARAMETER :: ls_d_cp = l_s / c_p !< precomputed l_s / c_p
101	REAL(wp), PARAMETER :: lv_d_rd = l_v / r_d !< precomputed l_v / r_d
102	REAL(wp), PARAMETER :: rd_d_rv = r_d / r_v !< precomputed r_d / r_v
103	REAL(wp), PARAMETER :: rd_d_cp = r_d / c_p !< precomputed r_d / c_p
104
105	REAL(wp) :: molecular_weight_of_solute = molecular_weight_of_nacl !< mol. m. NaCl (kg mol-1)
106	REAL(wp) :: rho_s = rho_nacl !< density of NaCl (kg m-3)
107	REAL(wp) :: vanthoff = vanthoff_nacl !< van't Hoff factor for NaCl
108
109	SAVE
110
111	PRIVATE magnus_0d, &
112	magnus_1d, &
113	magnus_tl_0d, &
114	magnus_tl_1d, &
115	magnus_0d_ice, &
116	magnus_1d_ice, &
117	ideal_gas_law_rho_0d, &
118	ideal_gas_law_rho_1d, &
119	ideal_gas_law_rho_pt_0d, &
120	ideal_gas_law_rho_pt_1d, &
121	exner_function_0d, &
122	exner_function_1d, &
123	exner_function_invers_0d, &
124	exner_function_invers_1d, &
125	barometric_formula_0d, &
126	barometric_formula_1d
127
128
129	INTERFACE convert_utm_to_geographic
130	MODULE PROCEDURE convert_utm_to_geographic
131	END INTERFACE convert_utm_to_geographic
132
133	INTERFACE magnus
134	MODULE PROCEDURE magnus_0d
135	MODULE PROCEDURE magnus_1d
136	END INTERFACE magnus
137
138	INTERFACE magnus_tl
139	MODULE PROCEDURE magnus_tl_0d
140	MODULE PROCEDURE magnus_tl_1d
141	END INTERFACE magnus_tl
142
143	INTERFACE magnus_ice
144	MODULE PROCEDURE magnus_0d_ice
145	MODULE PROCEDURE magnus_1d_ice
146	END INTERFACE magnus_ice
147
148	INTERFACE ideal_gas_law_rho
149	MODULE PROCEDURE ideal_gas_law_rho_0d
150	MODULE PROCEDURE ideal_gas_law_rho_1d
151	END INTERFACE ideal_gas_law_rho
152
153	INTERFACE ideal_gas_law_rho_pt
154	MODULE PROCEDURE ideal_gas_law_rho_pt_0d
155	MODULE PROCEDURE ideal_gas_law_rho_pt_1d
156	END INTERFACE ideal_gas_law_rho_pt
157
158	INTERFACE exner_function
159	MODULE PROCEDURE exner_function_0d
160	MODULE PROCEDURE exner_function_1d
161	END INTERFACE exner_function
162
163	INTERFACE exner_function_invers
164	MODULE PROCEDURE exner_function_invers_0d
165	MODULE PROCEDURE exner_function_invers_1d
166	END INTERFACE exner_function_invers
167
168	INTERFACE barometric_formula
169	MODULE PROCEDURE barometric_formula_0d
170	MODULE PROCEDURE barometric_formula_1d
171	END INTERFACE barometric_formula
172	!
173	!-- Public routines
174	PUBLIC convert_utm_to_geographic
175
176	CONTAINS
177
178
179	!--------------------------------------------------------------------------------------------------!
180	! Description:
181	! ------------
182	!> Convert UTM coordinates into geographic latitude and longitude. Conversion is based on the work
183	!> of KrÃŒger (1912) DOI: 10.2312/GFZ.b103-krueger28 and Karney (2013) DOI: 10.1007/s00190-012-0578-z
184	!> Based on a JavaScript of the geodesy function library written by chrisveness
185	!> https://github.com/chrisveness/geodesy
186	!--------------------------------------------------------------------------------------------------!
187	SUBROUTINE convert_utm_to_geographic( crs, eutm, nutm, lon, lat )
188
189	INTEGER(iwp) :: j !< loop index
190
191	REAL(wp), INTENT(in) :: eutm !< easting (UTM)
192	REAL(wp), INTENT(out) :: lat !< geographic latitude in degree
193	REAL(wp), INTENT(out) :: lon !< geographic longitude in degree
194	REAL(wp), INTENT(in) :: nutm !< northing (UTM)
195
196	REAL(wp) :: a !< 2pia is the circumference of a meridian
197	REAL(wp) :: cos_eta_s !< cos(eta_s)
198	REAL(wp) :: delta_i !<
199	REAL(wp) :: delta_tau_i !<
200	REAL(wp) :: e !< eccentricity
201	REAL(wp) :: eta !<
202	REAL(wp) :: eta_s !<
203	REAL(wp) :: n !< 3rd flattening
204	REAL(wp) :: n2 !< n^2
205	REAL(wp) :: n3 !< n^3
206	REAL(wp) :: n4 !< n^4
207	REAL(wp) :: n5 !< n^5
208	REAL(wp) :: n6 !< n^6
209	REAL(wp) :: nu !<
210	REAL(wp) :: nu_s !<
211	REAL(wp) :: sin_eta_s !< sin(eta_s)
212	REAL(wp) :: sinh_nu_s !< sinush(nu_s)
213	REAL(wp) :: tau_i !<
214	REAL(wp) :: tau_i_s !<
215	REAL(wp) :: tau_s !<
216	REAL(wp) :: x !< adjusted easting
217	REAL(wp) :: y !< adjusted northing
218
219	REAL(wp), DIMENSION(6) :: beta !< 6th order KrÃŒger expressions
220
221	REAL(wp), DIMENSION(8), INTENT(in) :: crs !< coordinate reference system, consists of
222	!< (/semi_major_axis,
223	!< inverse_flattening,
224	!< longitude_of_prime_meridian,
225	!< longitude_of_central_meridian,
226	!< scale_factor_at_central_meridian,
227	!< latitude_of_projection_origin,
228	!< false_easting,
229	!< false_northing /)
230
231	x = eutm - crs(7) ! remove false easting
232	y = nutm - crs(8) ! remove false northing
233	!
234	!-- From Karney 2011 Eq 15-22, 36:
235	e = SQRT( 1.0_wp / crs(2) * ( 2.0_wp - 1.0_wp / crs(2) ) )
236	n = 1.0_wp / crs(2) / ( 2.0_wp - 1.0_wp / crs(2) )
237	n2 = n * n
238	n3 = n * n2
239	n4 = n * n3
240	n5 = n * n4
241	n6 = n * n5
242
243	a = crs(1) / ( 1.0_wp + n ) * ( 1.0_wp + 0.25_wp * n2 + 0.015625_wp * n4 + 3.90625E-3_wp * n6 )
244
245	nu = x / ( crs(5) * a )
246	eta = y / ( crs(5) * a )
247
248	!-- According to KrÃŒger (1912), eq. 26*
249	beta(1) = 0.5_wp * n &
250	- 2.0_wp / 3.0_wp * n2 &
251	+ 37.0_wp / 96.0_wp * n3 &
252	- 1.0_wp / 360.0_wp * n4 &
253	- 81.0_wp / 512.0_wp * n5 &
254	+ 96199.0_wp / 604800.0_wp * n6
255
256	beta(2) = 1.0_wp / 48.0_wp * n2 &
257	+ 1.0_wp / 15.0_wp * n3 &
258	- 437.0_wp / 1440.0_wp * n4 &
259	+ 46.0_wp / 105.0_wp * n5 &
260	- 1118711.0_wp / 3870720.0_wp * n6
261
262	beta(3) = 17.0_wp / 480.0_wp * n3 &
263	- 37.0_wp / 840.0_wp * n4 &
264	- 209.0_wp / 4480.0_wp * n5 &
265	+ 5569.0_wp / 90720.0_wp * n6
266
267	beta(4) = 4397.0_wp / 161280.0_wp * n4 &
268	- 11.0_wp / 504.0_wp * n5 &
269	- 830251.0_wp / 7257600.0_wp * n6
270
271	beta(5) = 4583.0_wp / 161280.0_wp * n5 &
272	- 108847.0_wp / 3991680.0_wp * n6
273
274	beta(6) = 20648693.0_wp / 638668800.0_wp * n6
275
276	eta_s = eta
277	nu_s = nu
278	DO j = 1, 6
279	eta_s = eta_s - beta(j) * SIN(2.0_wp * j * eta) * COSH(2.0_wp * j * nu)
280	nu_s = nu_s - beta(j) * COS(2.0_wp * j * eta) * SINH(2.0_wp * j * nu)
281	ENDDO
282
283	sinh_nu_s = SINH( nu_s )
284	sin_eta_s = SIN( eta_s )
285	cos_eta_s = COS( eta_s )
286
287	tau_s = sin_eta_s / SQRT( sinh_nu_s2 + cos_eta_s2 )
288
289	tau_i = tau_s
290	delta_tau_i = 1.0_wp
291
292	DO WHILE ( ABS( delta_tau_i ) > 1.0E-12_wp )
293
294	delta_i = SINH( e * ATANH( e * tau_i / SQRT( 1.0_wp + tau_i**2 ) ) )
295
296	tau_i_s = tau_i * SQRT( 1.0_wp + delta_i*2 ) - delta_i SQRT( 1.0_wp + tau_i**2 )
297
298	delta_tau_i = ( tau_s - tau_i_s ) / SQRT( 1.0_wp + tau_i_s**2 ) &
299	* ( 1.0_wp + ( 1.0_wp - e*2 ) tau_i**2 ) &
300	/ ( ( 1.0_wp - e*2 ) SQRT( 1.0_wp + tau_i**2 ) )
301
302	tau_i = tau_i + delta_tau_i
303
304	ENDDO
305
306	lat = ATAN( tau_i ) / pi * 180.0_wp
307	lon = ATAN2( sinh_nu_s, cos_eta_s ) / pi * 180.0_wp + crs(4)
308
309	END SUBROUTINE convert_utm_to_geographic
310
311	!--------------------------------------------------------------------------------------------------!
312	! Description:
313	! ------------
314	!> This function computes the magnus formula (Press et al., 1992).
315	!> The magnus formula is needed to calculate the saturation vapor pressure.
316	!--------------------------------------------------------------------------------------------------!
317	FUNCTION magnus_0d( t )
318
319	IMPLICIT NONE
320
321	REAL(wp), INTENT(IN) :: t !< temperature (K)
322
323	REAL(wp) :: magnus_0d
324
325	!
326	!-- Saturation vapor pressure for a specific temperature:
327	magnus_0d = 611.2_wp * EXP( 17.62_wp * ( t - degc_to_k ) / ( t - 29.65_wp ) )
328
329	END FUNCTION magnus_0d
330
331	!--------------------------------------------------------------------------------------------------!
332	! Description:
333	! ------------
334	!> This function computes the magnus formula (Press et al., 1992).
335	!> The magnus formula is needed to calculate the saturation vapor pressure.
336	!--------------------------------------------------------------------------------------------------!
337	FUNCTION magnus_1d( t )
338
339	IMPLICIT NONE
340
341	REAL(wp), INTENT(IN), DIMENSION(:) :: t !< temperature (K)
342
343	REAL(wp), DIMENSION(size(t)) :: magnus_1d
344
345	!
346	!-- Saturation vapor pressure for a specific temperature:
347	magnus_1d = 611.2_wp * EXP( 17.62_wp * ( t - degc_to_k ) / ( t - 29.65_wp ) )
348
349	END FUNCTION magnus_1d
350
351	!--------------------------------------------------------------------------------------------------!
352	! Description:
353	! ------------
354	!> This function computes the magnus formula (Press et al., 1992) using the (ice-) liquid water
355	!> potential temperature.
356	!> The magnus formula is needed to calculate the saturation vapor pressure over a plane liquid water
357	!> surface.
358	!--------------------------------------------------------------------------------------------------!
359	FUNCTION magnus_tl_0d( t_l )
360
361	IMPLICIT NONE
362
363	REAL(wp), INTENT(IN) :: t_l !< liquid water temperature (K)
364
365	REAL(wp) :: magnus_tl_0d
366
367	!
368	!-- Saturation vapor pressure for a specific temperature:
369	magnus_tl_0d = 610.78_wp * EXP( 17.269_wp * ( t_l - 273.16_wp ) / ( t_l - 35.86_wp ) )
370
371	END FUNCTION magnus_tl_0d
372
373	!--------------------------------------------------------------------------------------------------!
374	! Description:
375	! ------------
376	!> This function computes the magnus formula (Press et al., 1992) using the (ice-) liquid water
377	!> potential temperature.
378	!> The magnus formula is needed to calculate the saturation vapor pressure over a plane liquid water
379	!> surface.
380	!--------------------------------------------------------------------------------------------------!
381	FUNCTION magnus_tl_1d( t_l )
382
383	IMPLICIT NONE
384
385	REAL(wp), INTENT(IN), DIMENSION(:) :: t_l !< liquid water temperature (K)
386
387	REAL(wp), DIMENSION(size(t_l)) :: magnus_tl_1d
388	!
389	!-- Saturation vapor pressure for a specific temperature:
390	magnus_tl_1d = 610.78_wp * EXP( 17.269_wp * ( t_l - 273.16_wp ) / ( t_l - 35.86_wp ) )
391
392	END FUNCTION magnus_tl_1d
393
394	!--------------------------------------------------------------------------------------------------!
395	! Description:
396	! ------------
397	!> This function computes the magnus formula (Press et al., 1992).
398	!> The magnus formula is needed to calculate the saturation vapor pressure over a plane ice surface.
399	!--------------------------------------------------------------------------------------------------!
400	FUNCTION magnus_0d_ice( t )
401
402	IMPLICIT NONE
403
404	REAL(wp), INTENT(IN) :: t !< temperature (K)
405
406	REAL(wp) :: magnus_0d_ice
407
408	!
409	!-- Saturation vapor pressure for a specific temperature:
410	magnus_0d_ice = 611.2_wp * EXP( 22.46_wp * ( t - degc_to_k ) / ( t - 0.53_wp ) )
411
412	END FUNCTION magnus_0d_ice
413
414	!--------------------------------------------------------------------------------------------------!
415	! Description:
416	! ------------
417	!> This function computes the magnus formula (Press et al., 1992).
418	!> The magnus formula is needed to calculate the saturation vapor pressure over a plane ice surface.
419	!--------------------------------------------------------------------------------------------------!
420	FUNCTION magnus_1d_ice( t )
421
422	IMPLICIT NONE
423
424	REAL(wp), INTENT(IN), DIMENSION(:) :: t !< temperature (K)
425
426	REAL(wp), DIMENSION(size(t)) :: magnus_1d_ice
427
428	!
429	!-- Saturation vapor pressure for a specific temperature:
430	magnus_1d_ice = 611.2_wp * EXP( 22.46_wp * ( t - degc_to_k ) / ( t - 0.53_wp ) )
431
432	END FUNCTION magnus_1d_ice
433
434	!--------------------------------------------------------------------------------------------------!
435	! Description:
436	! ------------
437	!> Compute the ideal gas law for scalar arguments.
438	!--------------------------------------------------------------------------------------------------!
439	FUNCTION ideal_gas_law_rho_0d( p, t )
440
441	IMPLICIT NONE
442
443	REAL(wp), INTENT(IN) :: p !< pressure (Pa)
444	REAL(wp), INTENT(IN) :: t !< temperature (K)
445
446	REAL(wp) :: ideal_gas_law_rho_0d
447
448	!
449	!-- Compute density according to ideal gas law:
450	ideal_gas_law_rho_0d = p / (r_d * t)
451
452	END FUNCTION ideal_gas_law_rho_0d
453
454	!--------------------------------------------------------------------------------------------------!
455	! Description:
456	! ------------
457	!> Compute the ideal gas law for 1-D array arguments.
458	!--------------------------------------------------------------------------------------------------!
459	FUNCTION ideal_gas_law_rho_1d( p, t )
460
461	IMPLICIT NONE
462
463	REAL(wp), INTENT(IN), DIMENSION(:) :: p !< pressure (Pa)
464	REAL(wp), INTENT(IN), DIMENSION(:) :: t !< temperature (K)
465
466	REAL(wp), DIMENSION(size(p)) :: ideal_gas_law_rho_1d
467
468	!
469	!-- Compute density according to ideal gas law:
470	ideal_gas_law_rho_1d = p / (r_d * t)
471
472	END FUNCTION ideal_gas_law_rho_1d
473
474	!--------------------------------------------------------------------------------------------------!
475	! Description:
476	! ------------
477	!> Compute the ideal gas law for scalar arguments.
478	!--------------------------------------------------------------------------------------------------!
479	FUNCTION ideal_gas_law_rho_pt_0d( p, t )
480
481	IMPLICIT NONE
482
483	REAL(wp), INTENT(IN) :: p !< pressure (Pa)
484	REAL(wp), INTENT(IN) :: t !< temperature (K)
485
486	REAL(wp) :: ideal_gas_law_rho_pt_0d
487
488	!
489	!-- Compute density according to ideal gas law:
490	ideal_gas_law_rho_pt_0d = p / (r_d * exner_function(p) * t)
491
492	END FUNCTION ideal_gas_law_rho_pt_0d
493
494	!--------------------------------------------------------------------------------------------------!
495	! Description:
496	! ------------
497	!> Compute the ideal gas law for 1-D array arguments.
498	!--------------------------------------------------------------------------------------------------!
499	FUNCTION ideal_gas_law_rho_pt_1d( p, t )
500
501	IMPLICIT NONE
502
503	REAL(wp), INTENT(IN), DIMENSION(:) :: p !< pressure (Pa)
504	REAL(wp), INTENT(IN), DIMENSION(:) :: t !< temperature (K)
505
506	REAL(wp), DIMENSION(size(p)) :: ideal_gas_law_rho_pt_1d
507
508	!
509	!-- Compute density according to ideal gas law:
510	ideal_gas_law_rho_pt_1d = p / (r_d * exner_function(p) * t)
511
512	END FUNCTION ideal_gas_law_rho_pt_1d
513
514	!--------------------------------------------------------------------------------------------------!
515	! Description:
516	! ------------
517	!> Compute the exner function for scalar arguments.
518	!--------------------------------------------------------------------------------------------------!
519	FUNCTION exner_function_0d( p )
520
521	IMPLICIT NONE
522
523	REAL(wp), INTENT(IN) :: p !< pressure (Pa)
524
525	REAL(wp) :: exner_function_0d
526
527	!
528	!-- Compute exner function:
529	exner_function_0d = ( p / p_0 )**( rd_d_cp )
530
531	END FUNCTION exner_function_0d
532
533	!--------------------------------------------------------------------------------------------------!
534	! Description:
535	! ------------
536	!> Compute the exner function for 1-D array arguments.
537	!--------------------------------------------------------------------------------------------------!
538	FUNCTION exner_function_1d( p )
539
540	IMPLICIT NONE
541
542	REAL(wp), INTENT(IN), DIMENSION(:) :: p !< pressure (Pa)
543
544	REAL(wp), DIMENSION(size(p)) :: exner_function_1d
545
546	!
547	!-- Compute exner function:
548	exner_function_1d = ( p / p_0 )**( rd_d_cp )
549
550	END FUNCTION exner_function_1d
551
552	!--------------------------------------------------------------------------------------------------!
553	! Description:
554	! ------------
555	!> Compute the exner function for scalar arguments.
556	!--------------------------------------------------------------------------------------------------!
557	FUNCTION exner_function_invers_0d( p )
558
559	IMPLICIT NONE
560
561	REAL(wp), INTENT(IN) :: p !< pressure (Pa)
562
563	REAL(wp) :: exner_function_invers_0d
564
565	!
566	!-- Compute exner function:
567	exner_function_invers_0d = ( p_0 / p )**( rd_d_cp )
568
569	END FUNCTION exner_function_invers_0d
570
571	!--------------------------------------------------------------------------------------------------!
572	! Description:
573	! ------------
574	!> Compute the exner function for 1-D array arguments.
575	!--------------------------------------------------------------------------------------------------!
576	FUNCTION exner_function_invers_1d( p )
577
578	IMPLICIT NONE
579
580	REAL(wp), INTENT(IN), DIMENSION(:) :: p !< pressure (Pa)
581
582	REAL(wp), DIMENSION(size(p)) :: exner_function_invers_1d
583
584	!
585	!-- Compute exner function:
586	exner_function_invers_1d = ( p_0 / p )**( rd_d_cp )
587
588	END FUNCTION exner_function_invers_1d
589
590	!--------------------------------------------------------------------------------------------------!
591	! Description:
592	! ------------
593	!> Compute the barometric formula for scalar arguments. The calculation is based on the assumption
594	!> of a polytropic atmosphere and neutral stratification, where the temperature lapse rate is g/cp.
595	!--------------------------------------------------------------------------------------------------!
596	FUNCTION barometric_formula_0d( z, t_0, p_0)
597
598	IMPLICIT NONE
599
600	REAL(wp), INTENT(IN) :: z !< height (m)
601	REAL(wp), INTENT(IN) :: t_0 !< temperature reference state (K)
602	REAL(wp), INTENT(IN) :: p_0 !< surface pressure (Pa)
603
604	REAL(wp) :: barometric_formula_0d
605
606	!
607	!-- Compute barometric formula:
608	barometric_formula_0d = p_0 * ( (t_0 - g_d_cp * z) / t_0 )**( cp_d_rd )
609
610	END FUNCTION barometric_formula_0d
611
612	!--------------------------------------------------------------------------------------------------!
613	! Description:
614	! ------------
615	!> Compute the barometric formula for 1-D array arguments. The calculation is based on the
616	!> assumption of a polytropic atmosphere and neutral stratification, where the temperature lapse
617	!> rate is g/cp.
618	!--------------------------------------------------------------------------------------------------!
619	FUNCTION barometric_formula_1d( z, t_0, p_0)
620
621	IMPLICIT NONE
622
623	REAL(wp), INTENT(IN), DIMENSION(:) :: z !< height (m)
624	REAL(wp), INTENT(IN) :: t_0 !< temperature reference state (K)
625	REAL(wp), INTENT(IN) :: p_0 !< surface pressure (Pa)
626
627	REAL(wp), DIMENSION(size(z)) :: barometric_formula_1d
628
629	!
630	!-- Compute barometric formula:
631	barometric_formula_1d = p_0 * ( (t_0 - g_d_cp * z) / t_0 )**( cp_d_rd )
632
633	END FUNCTION barometric_formula_1d
634
635	END MODULE basic_constants_and_equations_mod

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