| 1 | """Class Domain - |
|---|
| 2 | 1D interval domains for finite-volume computations of |
|---|
| 3 | the shallow water wave equation. |
|---|
| 4 | |
|---|
| 5 | This module contains a specialisation of class Domain from module domain.py |
|---|
| 6 | consisting of methods specific to the Shallow Water Wave Equation |
|---|
| 7 | |
|---|
| 8 | |
|---|
| 9 | U_t + E_x = S |
|---|
| 10 | |
|---|
| 11 | where |
|---|
| 12 | |
|---|
| 13 | U = [w, uh] |
|---|
| 14 | E = [uh, u^2h + gh^2/2] |
|---|
| 15 | S represents source terms forcing the system |
|---|
| 16 | (e.g. gravity, friction, wind stress, ...) |
|---|
| 17 | |
|---|
| 18 | and _t, _x, _y denote the derivative with respect to t, x and y respectively. |
|---|
| 19 | |
|---|
| 20 | The quantities are |
|---|
| 21 | |
|---|
| 22 | symbol variable name explanation |
|---|
| 23 | x x horizontal distance from origin [m] |
|---|
| 24 | z elevation elevation of bed on which flow is modelled [m] |
|---|
| 25 | h height water height above z [m] |
|---|
| 26 | w stage absolute water level, w = z+h [m] |
|---|
| 27 | u speed in the x direction [m/s] |
|---|
| 28 | uh xmomentum momentum in the x direction [m^2/s] |
|---|
| 29 | |
|---|
| 30 | eta mannings friction coefficient [to appear] |
|---|
| 31 | nu wind stress coefficient [to appear] |
|---|
| 32 | |
|---|
| 33 | The conserved quantities are w, uh |
|---|
| 34 | |
|---|
| 35 | For details see e.g. |
|---|
| 36 | Christopher Zoppou and Stephen Roberts, |
|---|
| 37 | Catastrophic Collapse of Water Supply Reservoirs in Urban Areas, |
|---|
| 38 | Journal of Hydraulic Engineering, vol. 127, No. 7 July 1999 |
|---|
| 39 | |
|---|
| 40 | |
|---|
| 41 | John Jakeman, Ole Nielsen, Stephen Roberts, Duncan Gray, Christopher Zoppou |
|---|
| 42 | Geoscience Australia, 2006 |
|---|
| 43 | """ |
|---|
| 44 | |
|---|
| 45 | from domain import * |
|---|
| 46 | Generic_Domain = Domain #Rename |
|---|
| 47 | |
|---|
| 48 | #Shallow water domain |
|---|
| 49 | class Domain(Generic_Domain): |
|---|
| 50 | |
|---|
| 51 | def __init__(self, coordinates, boundary = None, tagged_elements = None, |
|---|
| 52 | geo_reference = None): |
|---|
| 53 | |
|---|
| 54 | conserved_quantities = ['stage', 'xmomentum'] |
|---|
| 55 | other_quantities = ['elevation', 'friction'] |
|---|
| 56 | Generic_Domain.__init__(self, coordinates, boundary, |
|---|
| 57 | conserved_quantities, other_quantities, |
|---|
| 58 | tagged_elements, geo_reference) |
|---|
| 59 | |
|---|
| 60 | from config import minimum_allowed_height, g |
|---|
| 61 | self.minimum_allowed_height = minimum_allowed_height |
|---|
| 62 | self.g = g |
|---|
| 63 | |
|---|
| 64 | #forcing terms not included in 1d domain ?WHy? |
|---|
| 65 | self.forcing_terms.append(gravity) |
|---|
| 66 | #self.forcing_terms.append(manning_friction) |
|---|
| 67 | #print "\nI have Removed forcing terms line 64 1dsw" |
|---|
| 68 | |
|---|
| 69 | #Realtime visualisation |
|---|
| 70 | self.visualiser = None |
|---|
| 71 | self.visualise = False |
|---|
| 72 | self.visualise_color_stage = False |
|---|
| 73 | self.visualise_stage_range = 1.0 |
|---|
| 74 | self.visualise_timer = True |
|---|
| 75 | self.visualise_range_z = None |
|---|
| 76 | |
|---|
| 77 | #Stored output |
|---|
| 78 | self.store = True |
|---|
| 79 | self.format = 'sww' |
|---|
| 80 | self.smooth = True |
|---|
| 81 | |
|---|
| 82 | #Evolve parametrs |
|---|
| 83 | self.cfl = 1.0 |
|---|
| 84 | |
|---|
| 85 | #Reduction operation for get_vertex_values |
|---|
| 86 | from util import mean |
|---|
| 87 | self.reduction = mean |
|---|
| 88 | #self.reduction = min #Looks better near steep slopes |
|---|
| 89 | |
|---|
| 90 | self.quantities_to_be_stored = ['stage','xmomentum'] |
|---|
| 91 | |
|---|
| 92 | def set_quantities_to_be_stored(self, q): |
|---|
| 93 | """Specify which quantities will be stored in the sww file. |
|---|
| 94 | |
|---|
| 95 | q must be either: |
|---|
| 96 | - the name of a quantity |
|---|
| 97 | - a list of quantity names |
|---|
| 98 | - None |
|---|
| 99 | |
|---|
| 100 | In the two first cases, the named quantities will be stored at each |
|---|
| 101 | yieldstep |
|---|
| 102 | (This is in addition to the quantities elevation and friction) |
|---|
| 103 | If q is None, storage will be switched off altogether. |
|---|
| 104 | """ |
|---|
| 105 | |
|---|
| 106 | |
|---|
| 107 | if q is None: |
|---|
| 108 | self.quantities_to_be_stored = [] |
|---|
| 109 | self.store = False |
|---|
| 110 | return |
|---|
| 111 | |
|---|
| 112 | if isinstance(q, basestring): |
|---|
| 113 | q = [q] # Turn argument into a list |
|---|
| 114 | |
|---|
| 115 | #Check correcness |
|---|
| 116 | for quantity_name in q: |
|---|
| 117 | msg = 'Quantity %s is not a valid conserved quantity' %quantity_name |
|---|
| 118 | assert quantity_name in self.conserved_quantities, msg |
|---|
| 119 | |
|---|
| 120 | self.quantities_to_be_stored = q |
|---|
| 121 | |
|---|
| 122 | |
|---|
| 123 | def initialise_visualiser(self,scale_z=1.0,rect=None): |
|---|
| 124 | #Realtime visualisation |
|---|
| 125 | if self.visualiser is None: |
|---|
| 126 | from realtime_visualisation_new import Visualiser |
|---|
| 127 | self.visualiser = Visualiser(self,scale_z,rect) |
|---|
| 128 | self.visualiser.setup['elevation']=True |
|---|
| 129 | self.visualiser.updating['stage']=True |
|---|
| 130 | self.visualise = True |
|---|
| 131 | if self.visualise_color_stage == True: |
|---|
| 132 | self.visualiser.coloring['stage'] = True |
|---|
| 133 | self.visualiser.qcolor['stage'] = (0.0, 0.0, 0.8) |
|---|
| 134 | print 'initialise visualiser' |
|---|
| 135 | print self.visualiser.setup |
|---|
| 136 | print self.visualiser.updating |
|---|
| 137 | |
|---|
| 138 | def check_integrity(self): |
|---|
| 139 | Generic_Domain.check_integrity(self) |
|---|
| 140 | #Check that we are solving the shallow water wave equation |
|---|
| 141 | |
|---|
| 142 | msg = 'First conserved quantity must be "stage"' |
|---|
| 143 | assert self.conserved_quantities[0] == 'stage', msg |
|---|
| 144 | msg = 'Second conserved quantity must be "xmomentum"' |
|---|
| 145 | assert self.conserved_quantities[1] == 'xmomentum', msg |
|---|
| 146 | |
|---|
| 147 | def extrapolate_second_order_sw(self): |
|---|
| 148 | #Call correct module function |
|---|
| 149 | #(either from this module or C-extension) |
|---|
| 150 | extrapolate_second_order_sw(self) |
|---|
| 151 | |
|---|
| 152 | def compute_fluxes(self): |
|---|
| 153 | #Call correct module function |
|---|
| 154 | #(either from this module or C-extension) |
|---|
| 155 | compute_fluxes(self) |
|---|
| 156 | |
|---|
| 157 | def distribute_to_vertices_and_edges(self): |
|---|
| 158 | #Call correct module function |
|---|
| 159 | #(either from this module or C-extension) |
|---|
| 160 | distribute_to_vertices_and_edges(self) |
|---|
| 161 | |
|---|
| 162 | def evolve(self, yieldstep = None, finaltime = None, |
|---|
| 163 | skip_initial_step = False): |
|---|
| 164 | """Specialisation of basic evolve method from parent class |
|---|
| 165 | """ |
|---|
| 166 | |
|---|
| 167 | #Call check integrity here rather than from user scripts |
|---|
| 168 | #self.check_integrity() |
|---|
| 169 | |
|---|
| 170 | #msg = 'Parameter beta_h must be in the interval [0, 1)' |
|---|
| 171 | #assert 0 <= self.beta_h < 1.0, msg |
|---|
| 172 | #msg = 'Parameter beta_w must be in the interval [0, 1)' |
|---|
| 173 | #assert 0 <= self.beta_w < 1.0, msg |
|---|
| 174 | |
|---|
| 175 | |
|---|
| 176 | #Initial update of vertex and edge values before any storage |
|---|
| 177 | #and or visualisation |
|---|
| 178 | self.distribute_to_vertices_and_edges() |
|---|
| 179 | |
|---|
| 180 | #Initialise real time viz if requested |
|---|
| 181 | #if self.visualise is True and self.time == 0.0: |
|---|
| 182 | # if self.visualiser is None: |
|---|
| 183 | # self.initialise_visualiser() |
|---|
| 184 | # |
|---|
| 185 | # self.visualiser.update_timer() |
|---|
| 186 | # self.visualiser.setup_all() |
|---|
| 187 | |
|---|
| 188 | #Store model data, e.g. for visualisation |
|---|
| 189 | #if self.store is True and self.time == 0.0: |
|---|
| 190 | # self.initialise_storage() |
|---|
| 191 | # #print 'Storing results in ' + self.writer.filename |
|---|
| 192 | #else: |
|---|
| 193 | # pass |
|---|
| 194 | # #print 'Results will not be stored.' |
|---|
| 195 | # #print 'To store results set domain.store = True' |
|---|
| 196 | # #FIXME: Diagnostic output should be controlled by |
|---|
| 197 | # # a 'verbose' flag living in domain (or in a parent class) |
|---|
| 198 | |
|---|
| 199 | #Call basic machinery from parent class |
|---|
| 200 | for t in Generic_Domain.evolve(self, yieldstep, finaltime, |
|---|
| 201 | skip_initial_step): |
|---|
| 202 | #Real time viz |
|---|
| 203 | # if self.visualise is True: |
|---|
| 204 | # self.visualiser.update_all() |
|---|
| 205 | # self.visualiser.update_timer() |
|---|
| 206 | |
|---|
| 207 | |
|---|
| 208 | #Store model data, e.g. for subsequent visualisation |
|---|
| 209 | # if self.store is True: |
|---|
| 210 | # self.store_timestep(self.quantities_to_be_stored) |
|---|
| 211 | |
|---|
| 212 | #FIXME: Could maybe be taken from specified list |
|---|
| 213 | #of 'store every step' quantities |
|---|
| 214 | |
|---|
| 215 | #Pass control on to outer loop for more specific actions |
|---|
| 216 | yield(t) |
|---|
| 217 | |
|---|
| 218 | def initialise_storage(self): |
|---|
| 219 | """Create and initialise self.writer object for storing data. |
|---|
| 220 | Also, save x and bed elevation |
|---|
| 221 | """ |
|---|
| 222 | |
|---|
| 223 | import data_manager |
|---|
| 224 | |
|---|
| 225 | #Initialise writer |
|---|
| 226 | self.writer = data_manager.get_dataobject(self, mode = 'w') |
|---|
| 227 | |
|---|
| 228 | #Store vertices and connectivity |
|---|
| 229 | self.writer.store_connectivity() |
|---|
| 230 | |
|---|
| 231 | |
|---|
| 232 | def store_timestep(self, name): |
|---|
| 233 | """Store named quantity and time. |
|---|
| 234 | |
|---|
| 235 | Precondition: |
|---|
| 236 | self.write has been initialised |
|---|
| 237 | """ |
|---|
| 238 | self.writer.store_timestep(name) |
|---|
| 239 | |
|---|
| 240 | |
|---|
| 241 | #=============== End of Shallow Water Domain =============================== |
|---|
| 242 | |
|---|
| 243 | #Rotation of momentum vector |
|---|
| 244 | def rotate(q, normal, direction = 1): |
|---|
| 245 | """Rotate the momentum component q (q[1], q[2]) |
|---|
| 246 | from x,y coordinates to coordinates based on normal vector. |
|---|
| 247 | |
|---|
| 248 | If direction is negative the rotation is inverted. |
|---|
| 249 | |
|---|
| 250 | Input vector is preserved |
|---|
| 251 | |
|---|
| 252 | This function is specific to the shallow water wave equation |
|---|
| 253 | """ |
|---|
| 254 | |
|---|
| 255 | from Numeric import zeros, Float |
|---|
| 256 | |
|---|
| 257 | assert len(q) == 3,\ |
|---|
| 258 | 'Vector of conserved quantities must have length 3'\ |
|---|
| 259 | 'for 2D shallow water equation' |
|---|
| 260 | |
|---|
| 261 | try: |
|---|
| 262 | l = len(normal) |
|---|
| 263 | except: |
|---|
| 264 | raise 'Normal vector must be an Numeric array' |
|---|
| 265 | |
|---|
| 266 | assert l == 2, 'Normal vector must have 2 components' |
|---|
| 267 | |
|---|
| 268 | |
|---|
| 269 | n1 = normal[0] |
|---|
| 270 | n2 = normal[1] |
|---|
| 271 | |
|---|
| 272 | r = zeros(len(q), Float) #Rotated quantities |
|---|
| 273 | r[0] = q[0] #First quantity, height, is not rotated |
|---|
| 274 | |
|---|
| 275 | if direction == -1: |
|---|
| 276 | n2 = -n2 |
|---|
| 277 | |
|---|
| 278 | |
|---|
| 279 | r[1] = n1*q[1] + n2*q[2] |
|---|
| 280 | r[2] = -n2*q[1] + n1*q[2] |
|---|
| 281 | |
|---|
| 282 | return r |
|---|
| 283 | |
|---|
| 284 | |
|---|
| 285 | def flux_function_unsplit(normal, ql, qr, zl, zr): |
|---|
| 286 | """Compute fluxes between volumes for the shallow water wave equation |
|---|
| 287 | cast in terms of w = h+z using the 'central scheme' as described in |
|---|
| 288 | |
|---|
| 289 | Kurganov, Noelle, Petrova. 'Semidiscrete Central-Upwind Schemes For |
|---|
| 290 | Hyperbolic Conservation Laws and Hamilton-Jacobi Equations'. |
|---|
| 291 | Siam J. Sci. Comput. Vol. 23, No. 3, pp. 707-740. |
|---|
| 292 | |
|---|
| 293 | The implemented formula is given in equation (3.15) on page 714 |
|---|
| 294 | |
|---|
| 295 | Conserved quantities w, uh, are stored as elements 0 and 1 |
|---|
| 296 | in the numerical vectors ql an qr. |
|---|
| 297 | |
|---|
| 298 | Bed elevations zl and zr. |
|---|
| 299 | """ |
|---|
| 300 | |
|---|
| 301 | from config import g, epsilon |
|---|
| 302 | from math import sqrt |
|---|
| 303 | from Numeric import array |
|---|
| 304 | |
|---|
| 305 | #print 'ql',ql |
|---|
| 306 | |
|---|
| 307 | #Align momentums with x-axis |
|---|
| 308 | #q_left = rotate(ql, normal, direction = 1) |
|---|
| 309 | #q_right = rotate(qr, normal, direction = 1) |
|---|
| 310 | q_left = ql |
|---|
| 311 | q_left[1] = q_left[1]*normal |
|---|
| 312 | q_right = qr |
|---|
| 313 | q_right[1] = q_right[1]*normal |
|---|
| 314 | |
|---|
| 315 | #z = (zl+zr)/2 #Take average of field values |
|---|
| 316 | z = 0.5*(zl+zr) #Take average of field values |
|---|
| 317 | |
|---|
| 318 | w_left = q_left[0] #w=h+z |
|---|
| 319 | h_left = w_left-z |
|---|
| 320 | uh_left = q_left[1] |
|---|
| 321 | |
|---|
| 322 | if h_left < epsilon: |
|---|
| 323 | u_left = 0.0 #Could have been negative |
|---|
| 324 | h_left = 0.0 |
|---|
| 325 | else: |
|---|
| 326 | u_left = uh_left/h_left |
|---|
| 327 | |
|---|
| 328 | |
|---|
| 329 | w_right = q_right[0] #w=h+z |
|---|
| 330 | h_right = w_right-z |
|---|
| 331 | uh_right = q_right[1] |
|---|
| 332 | |
|---|
| 333 | |
|---|
| 334 | if h_right < epsilon: |
|---|
| 335 | u_right = 0.0 #Could have been negative |
|---|
| 336 | h_right = 0.0 |
|---|
| 337 | else: |
|---|
| 338 | u_right = uh_right/h_right |
|---|
| 339 | |
|---|
| 340 | #vh_left = q_left[2] |
|---|
| 341 | #vh_right = q_right[2] |
|---|
| 342 | |
|---|
| 343 | soundspeed_left = sqrt(g*h_left) |
|---|
| 344 | soundspeed_right = sqrt(g*h_right) |
|---|
| 345 | |
|---|
| 346 | #Maximal wave speed |
|---|
| 347 | s_max = max(u_left+soundspeed_left, u_right+soundspeed_right, 0) |
|---|
| 348 | |
|---|
| 349 | #Minimal wave speed |
|---|
| 350 | s_min = min(u_left-soundspeed_left, u_right-soundspeed_right, 0) |
|---|
| 351 | |
|---|
| 352 | #Flux computation |
|---|
| 353 | |
|---|
| 354 | #flux_left = array([u_left*h_left, |
|---|
| 355 | # u_left*uh_left + 0.5*g*h_left**2]) |
|---|
| 356 | #flux_right = array([u_right*h_right, |
|---|
| 357 | # u_right*uh_right + 0.5*g*h_right**2]) |
|---|
| 358 | flux_left = array([u_left*h_left, |
|---|
| 359 | u_left*uh_left + 0.5*g*h_left*h_left]) |
|---|
| 360 | flux_right = array([u_right*h_right, |
|---|
| 361 | u_right*uh_right + 0.5*g*h_right*h_right]) |
|---|
| 362 | |
|---|
| 363 | denom = s_max-s_min |
|---|
| 364 | if denom == 0.0: |
|---|
| 365 | edgeflux = array([0.0, 0.0]) |
|---|
| 366 | max_speed = 0.0 |
|---|
| 367 | else: |
|---|
| 368 | edgeflux = (s_max*flux_left - s_min*flux_right)/denom |
|---|
| 369 | edgeflux += s_max*s_min*(q_right-q_left)/denom |
|---|
| 370 | |
|---|
| 371 | edgeflux[1] = edgeflux[1]*normal |
|---|
| 372 | |
|---|
| 373 | max_speed = max(abs(s_max), abs(s_min)) |
|---|
| 374 | |
|---|
| 375 | return edgeflux, max_speed |
|---|
| 376 | |
|---|
| 377 | def flux_function_split(normal, ql, qr, zl, zr): |
|---|
| 378 | from config import g, epsilon |
|---|
| 379 | from math import sqrt |
|---|
| 380 | from Numeric import array |
|---|
| 381 | |
|---|
| 382 | #print 'ql',ql |
|---|
| 383 | |
|---|
| 384 | #Align momentums with x-axis |
|---|
| 385 | #q_left = rotate(ql, normal, direction = 1) |
|---|
| 386 | #q_right = rotate(qr, normal, direction = 1) |
|---|
| 387 | q_left = ql |
|---|
| 388 | q_left[1] = q_left[1]*normal |
|---|
| 389 | q_right = qr |
|---|
| 390 | q_right[1] = q_right[1]*normal |
|---|
| 391 | |
|---|
| 392 | #z = (zl+zr)/2 #Take average of field values |
|---|
| 393 | z = 0.5*(zl+zr) #Take average of field values |
|---|
| 394 | |
|---|
| 395 | w_left = q_left[0] #w=h+z |
|---|
| 396 | h_left = w_left-z |
|---|
| 397 | uh_left = q_left[1] |
|---|
| 398 | |
|---|
| 399 | if h_left < epsilon: |
|---|
| 400 | u_left = 0.0 #Could have been negative |
|---|
| 401 | h_left = 0.0 |
|---|
| 402 | else: |
|---|
| 403 | u_left = uh_left/h_left |
|---|
| 404 | |
|---|
| 405 | |
|---|
| 406 | w_right = q_right[0] #w=h+z |
|---|
| 407 | h_right = w_right-z |
|---|
| 408 | uh_right = q_right[1] |
|---|
| 409 | |
|---|
| 410 | |
|---|
| 411 | if h_right < epsilon: |
|---|
| 412 | u_right = 0.0 #Could have been negative |
|---|
| 413 | h_right = 0.0 |
|---|
| 414 | else: |
|---|
| 415 | u_right = uh_right/h_right |
|---|
| 416 | |
|---|
| 417 | #vh_left = q_left[2] |
|---|
| 418 | #vh_right = q_right[2] |
|---|
| 419 | |
|---|
| 420 | #soundspeed_left = sqrt(g*h_left) |
|---|
| 421 | #soundspeed_right = sqrt(g*h_right) |
|---|
| 422 | |
|---|
| 423 | #Maximal wave speed |
|---|
| 424 | #s_max = max(u_left+soundspeed_left, u_right+soundspeed_right, 0) |
|---|
| 425 | s_max = max(u_left, u_right, 0) |
|---|
| 426 | |
|---|
| 427 | #Minimal wave speed |
|---|
| 428 | #s_min = min(u_left-soundspeed_left, u_right-soundspeed_right, 0) |
|---|
| 429 | s_min = min(u_left, u_right, 0) |
|---|
| 430 | |
|---|
| 431 | #Flux computation |
|---|
| 432 | |
|---|
| 433 | #flux_left = array([u_left*h_left, |
|---|
| 434 | # u_left*uh_left + 0.5*g*h_left*h_left]) |
|---|
| 435 | #flux_right = array([u_right*h_right, |
|---|
| 436 | # u_right*uh_right + 0.5*g*h_right*h_right]) |
|---|
| 437 | flux_left = array([u_left*h_left, |
|---|
| 438 | u_left*uh_left])# + 0.5*g*h_left*h_left]) |
|---|
| 439 | flux_right = array([u_right*h_right, |
|---|
| 440 | u_right*uh_right])# + 0.5*g*h_right*h_right]) |
|---|
| 441 | |
|---|
| 442 | denom = s_max-s_min |
|---|
| 443 | if denom == 0.0: |
|---|
| 444 | edgeflux = array([0.0, 0.0]) |
|---|
| 445 | max_speed = 0.0 |
|---|
| 446 | else: |
|---|
| 447 | edgeflux = (s_max*flux_left - s_min*flux_right)/denom |
|---|
| 448 | edgeflux += s_max*s_min*(q_right-q_left)/denom |
|---|
| 449 | |
|---|
| 450 | edgeflux[1] = edgeflux[1]*normal |
|---|
| 451 | |
|---|
| 452 | max_speed = max(abs(s_max), abs(s_min)) |
|---|
| 453 | |
|---|
| 454 | return edgeflux, max_speed |
|---|
| 455 | |
|---|
| 456 | |
|---|
| 457 | def compute_fluxes(domain): |
|---|
| 458 | """Compute all fluxes and the timestep suitable for all volumes |
|---|
| 459 | in domain. |
|---|
| 460 | |
|---|
| 461 | Compute total flux for each conserved quantity using "flux_function" |
|---|
| 462 | |
|---|
| 463 | Fluxes across each edge are scaled by edgelengths and summed up |
|---|
| 464 | Resulting flux is then scaled by area and stored in |
|---|
| 465 | explicit_update for each of the three conserved quantities |
|---|
| 466 | stage, xmomentum and ymomentum |
|---|
| 467 | |
|---|
| 468 | The maximal allowable speed computed by the flux_function for each volume |
|---|
| 469 | is converted to a timestep that must not be exceeded. The minimum of |
|---|
| 470 | those is computed as the next overall timestep. |
|---|
| 471 | |
|---|
| 472 | Post conditions: |
|---|
| 473 | domain.explicit_update is reset to computed flux values |
|---|
| 474 | domain.timestep is set to the largest step satisfying all volumes. |
|---|
| 475 | """ |
|---|
| 476 | |
|---|
| 477 | import sys |
|---|
| 478 | from Numeric import zeros, Float |
|---|
| 479 | |
|---|
| 480 | N = domain.number_of_elements |
|---|
| 481 | |
|---|
| 482 | #Shortcuts |
|---|
| 483 | Stage = domain.quantities['stage'] |
|---|
| 484 | Xmom = domain.quantities['xmomentum'] |
|---|
| 485 | # Ymom = domain.quantities['ymomentum'] |
|---|
| 486 | Bed = domain.quantities['elevation'] |
|---|
| 487 | |
|---|
| 488 | #Arrays |
|---|
| 489 | #stage = Stage.edge_values |
|---|
| 490 | #xmom = Xmom.edge_values |
|---|
| 491 | # ymom = Ymom.edge_values |
|---|
| 492 | #bed = Bed.edge_values |
|---|
| 493 | |
|---|
| 494 | stage = Stage.vertex_values |
|---|
| 495 | xmom = Xmom.vertex_values |
|---|
| 496 | bed = Bed.vertex_values |
|---|
| 497 | |
|---|
| 498 | #print 'stage edge values', stage |
|---|
| 499 | #print 'xmom edge values', xmom |
|---|
| 500 | #print 'bed values', bed |
|---|
| 501 | |
|---|
| 502 | stage_bdry = Stage.boundary_values |
|---|
| 503 | xmom_bdry = Xmom.boundary_values |
|---|
| 504 | #print 'stage_bdry',stage_bdry |
|---|
| 505 | #print 'xmom_bdry', xmom_bdry |
|---|
| 506 | # ymom_bdry = Ymom.boundary_values |
|---|
| 507 | |
|---|
| 508 | # flux = zeros(3, Float) #Work array for summing up fluxes |
|---|
| 509 | flux = zeros(2, Float) #Work array for summing up fluxes |
|---|
| 510 | ql = zeros(2, Float) |
|---|
| 511 | qr = zeros(2, Float) |
|---|
| 512 | |
|---|
| 513 | #Loop |
|---|
| 514 | timestep = float(sys.maxint) |
|---|
| 515 | enter = True |
|---|
| 516 | for k in range(N): |
|---|
| 517 | |
|---|
| 518 | flux[:] = 0. #Reset work array |
|---|
| 519 | #for i in range(3): |
|---|
| 520 | for i in range(2): |
|---|
| 521 | #Quantities inside volume facing neighbour i |
|---|
| 522 | #ql[0] = stage[k, i] |
|---|
| 523 | #ql[1] = xmom[k, i] |
|---|
| 524 | ql = [stage[k, i], xmom[k, i]] |
|---|
| 525 | zl = bed[k, i] |
|---|
| 526 | |
|---|
| 527 | #Quantities at neighbour on nearest face |
|---|
| 528 | n = domain.neighbours[k,i] |
|---|
| 529 | if n < 0: |
|---|
| 530 | m = -n-1 #Convert negative flag to index |
|---|
| 531 | qr[0] = stage_bdry[m] |
|---|
| 532 | qr[1] = xmom_bdry[m] |
|---|
| 533 | zr = zl #Extend bed elevation to boundary |
|---|
| 534 | else: |
|---|
| 535 | #m = domain.neighbour_edges[k,i] |
|---|
| 536 | m = domain.neighbour_vertices[k,i] |
|---|
| 537 | #qr = [stage[n, m], xmom[n, m], ymom[n, m]] |
|---|
| 538 | qr[0] = stage[n, m] |
|---|
| 539 | qr[1] = xmom[n, m] |
|---|
| 540 | zr = bed[n, m] |
|---|
| 541 | |
|---|
| 542 | |
|---|
| 543 | #Outward pointing normal vector |
|---|
| 544 | normal = domain.normals[k, i] |
|---|
| 545 | |
|---|
| 546 | #Flux computation using provided function |
|---|
| 547 | #edgeflux, max_speed = flux_function(normal, ql, qr, zl, zr) |
|---|
| 548 | #print 'ql',ql |
|---|
| 549 | #print 'qr',qr |
|---|
| 550 | |
|---|
| 551 | |
|---|
| 552 | if domain.fluxfunc == "unsplit": |
|---|
| 553 | edgeflux, max_speed = flux_function_unsplit(normal, ql, qr, zl, zr) |
|---|
| 554 | elif domain.fluxfunc == "split": |
|---|
| 555 | edgeflux, max_speed = flux_function_split(normal, ql, qr, zl, zr) |
|---|
| 556 | #print 'edgeflux', edgeflux |
|---|
| 557 | |
|---|
| 558 | # THIS IS THE LINE TO DEAL WITH LEFT AND RIGHT FLUXES |
|---|
| 559 | # flux = edgefluxleft - edgefluxright |
|---|
| 560 | flux -= edgeflux #* domain.edgelengths[k,i] |
|---|
| 561 | #Update optimal_timestep |
|---|
| 562 | try: |
|---|
| 563 | #timestep = min(timestep, 0.5*domain.radii[k]/max_speed) |
|---|
| 564 | timestep = min(timestep, domain.cfl*0.5*domain.areas[k]/max_speed) |
|---|
| 565 | except ZeroDivisionError: |
|---|
| 566 | pass |
|---|
| 567 | |
|---|
| 568 | #Normalise by area and store for when all conserved |
|---|
| 569 | #quantities get updated |
|---|
| 570 | flux /= domain.areas[k] |
|---|
| 571 | |
|---|
| 572 | Stage.explicit_update[k] = flux[0] |
|---|
| 573 | Xmom.explicit_update[k] = flux[1] |
|---|
| 574 | #Ymom.explicit_update[k] = flux[2] |
|---|
| 575 | |
|---|
| 576 | |
|---|
| 577 | domain.timestep = timestep |
|---|
| 578 | |
|---|
| 579 | ## #see comments in the corresponding method in shallow_water_ext.c |
|---|
| 580 | ## def extrapolate_second_order_sw_c(domain): |
|---|
| 581 | ## """Wrapper calling C version of extrapolate_second_order_sw |
|---|
| 582 | ## """ |
|---|
| 583 | ## import sys |
|---|
| 584 | ## from Numeric import zeros, Float |
|---|
| 585 | |
|---|
| 586 | ## N = domain.number_of_elements |
|---|
| 587 | |
|---|
| 588 | ## #Shortcuts |
|---|
| 589 | ## Stage = domain.quantities['stage'] |
|---|
| 590 | ## Xmom = domain.quantities['xmomentum'] |
|---|
| 591 | ## # Ymom = domain.quantities['ymomentum'] |
|---|
| 592 | ## from shallow_water_ext import extrapolate_second_order_sw |
|---|
| 593 | ## extrapolate_second_order_sw(domain,domain.surrogate_neighbours, |
|---|
| 594 | ## # cant find this in domain domain.number_of_boundaries, |
|---|
| 595 | ## domain.centroid_coordinates, |
|---|
| 596 | ## Stage.centroid_values, |
|---|
| 597 | ## Xmom.centroid_values, |
|---|
| 598 | ## # Ymom.centroid_values, |
|---|
| 599 | ## domain.vertex_coordinates, |
|---|
| 600 | ## Stage.vertex_values, |
|---|
| 601 | ## Xmom.vertex_values)#, |
|---|
| 602 | ## # Ymom.vertex_values) |
|---|
| 603 | |
|---|
| 604 | ## def compute_fluxes_c(domain): |
|---|
| 605 | ## """Wrapper calling C version of compute fluxes |
|---|
| 606 | ## """ |
|---|
| 607 | |
|---|
| 608 | ## import sys |
|---|
| 609 | ## from Numeric import zeros, Float |
|---|
| 610 | |
|---|
| 611 | ## N = domain.number_of_elements |
|---|
| 612 | |
|---|
| 613 | ## #Shortcuts |
|---|
| 614 | ## Stage = domain.quantities['stage'] |
|---|
| 615 | ## Xmom = domain.quantities['xmomentum'] |
|---|
| 616 | ## #Ymom = domain.quantities['ymomentum'] |
|---|
| 617 | ## Bed = domain.quantities['elevation'] |
|---|
| 618 | |
|---|
| 619 | ## timestep = float(sys.maxint) |
|---|
| 620 | ## from shallow_water_ext import compute_fluxes |
|---|
| 621 | ## domain.timestep = compute_fluxes(timestep, domain.epsilon, domain.g, |
|---|
| 622 | ## domain.neighbours, |
|---|
| 623 | ## #domain.neighbour_edges, |
|---|
| 624 | ## domain.neighbour_vertices, |
|---|
| 625 | ## domain.normals, |
|---|
| 626 | ## #domain.edgelengths, |
|---|
| 627 | ## #domain.radii, |
|---|
| 628 | ## domain.areas, |
|---|
| 629 | ## #Stage.edge_values, |
|---|
| 630 | ## #Xmom.edge_values, |
|---|
| 631 | ## #Ymom.edge_values, |
|---|
| 632 | ## #Bed.edge_values, |
|---|
| 633 | ## Stage.vertex_values, |
|---|
| 634 | ## Xmom.vertex_values, |
|---|
| 635 | ## Bed.vertex_values, |
|---|
| 636 | ## Stage.boundary_values, |
|---|
| 637 | ## Xmom.boundary_values, |
|---|
| 638 | ## #Ymom.boundary_values, |
|---|
| 639 | ## Stage.explicit_update, |
|---|
| 640 | ## Xmom.explicit_update, |
|---|
| 641 | ## #Ymom.explicit_update, |
|---|
| 642 | ## domain.already_computed_flux) |
|---|
| 643 | |
|---|
| 644 | |
|---|
| 645 | #################################### |
|---|
| 646 | |
|---|
| 647 | # Module functions for gradient limiting (distribute_to_vertices_and_edges) |
|---|
| 648 | |
|---|
| 649 | def distribute_to_vertices_and_edges(domain): |
|---|
| 650 | """Distribution from centroids to vertices specific to the |
|---|
| 651 | shallow water wave |
|---|
| 652 | equation. |
|---|
| 653 | |
|---|
| 654 | It will ensure that h (w-z) is always non-negative even in the |
|---|
| 655 | presence of steep bed-slopes by taking a weighted average between shallow |
|---|
| 656 | and deep cases. |
|---|
| 657 | |
|---|
| 658 | In addition, all conserved quantities get distributed as per either a |
|---|
| 659 | constant (order==1) or a piecewise linear function (order==2). |
|---|
| 660 | |
|---|
| 661 | FIXME: more explanation about removal of artificial variability etc |
|---|
| 662 | |
|---|
| 663 | Precondition: |
|---|
| 664 | All quantities defined at centroids and bed elevation defined at |
|---|
| 665 | vertices. |
|---|
| 666 | |
|---|
| 667 | Postcondition |
|---|
| 668 | Conserved quantities defined at vertices |
|---|
| 669 | |
|---|
| 670 | """ |
|---|
| 671 | |
|---|
| 672 | #from config import optimised_gradient_limiter |
|---|
| 673 | |
|---|
| 674 | #Remove very thin layers of water |
|---|
| 675 | protect_against_infinitesimal_and_negative_heights(domain) |
|---|
| 676 | |
|---|
| 677 | #Extrapolate all conserved quantities |
|---|
| 678 | #if optimised_gradient_limiter: |
|---|
| 679 | # #MH090605 if second order, |
|---|
| 680 | # #perform the extrapolation and limiting on |
|---|
| 681 | # #all of the conserved quantities |
|---|
| 682 | |
|---|
| 683 | # if (domain.order == 1): |
|---|
| 684 | # for name in domain.conserved_quantities: |
|---|
| 685 | # Q = domain.quantities[name] |
|---|
| 686 | # Q.extrapolate_first_order() |
|---|
| 687 | # elif domain.order == 2: |
|---|
| 688 | # domain.extrapolate_second_order_sw() |
|---|
| 689 | # else: |
|---|
| 690 | # raise 'Unknown order' |
|---|
| 691 | #else: |
|---|
| 692 | #old code: |
|---|
| 693 | for name in domain.conserved_quantities: |
|---|
| 694 | Q = domain.quantities[name] |
|---|
| 695 | if domain.order == 1: |
|---|
| 696 | Q.extrapolate_first_order() |
|---|
| 697 | elif domain.order == 2: |
|---|
| 698 | Q.extrapolate_second_order() |
|---|
| 699 | #Q.limit() |
|---|
| 700 | else: |
|---|
| 701 | raise 'Unknown order' |
|---|
| 702 | |
|---|
| 703 | #Take bed elevation into account when water heights are small |
|---|
| 704 | #balance_deep_and_shallow(domain) |
|---|
| 705 | |
|---|
| 706 | #Compute edge values by interpolation |
|---|
| 707 | #for name in domain.conserved_quantities: |
|---|
| 708 | # Q = domain.quantities[name] |
|---|
| 709 | # Q.interpolate_from_vertices_to_edges() |
|---|
| 710 | |
|---|
| 711 | |
|---|
| 712 | def protect_against_infinitesimal_and_negative_heights(domain): |
|---|
| 713 | """Protect against infinitesimal heights and associated high velocities |
|---|
| 714 | """ |
|---|
| 715 | |
|---|
| 716 | #Shortcuts |
|---|
| 717 | wc = domain.quantities['stage'].centroid_values |
|---|
| 718 | zc = domain.quantities['elevation'].centroid_values |
|---|
| 719 | xmomc = domain.quantities['xmomentum'].centroid_values |
|---|
| 720 | # ymomc = domain.quantities['ymomentum'].centroid_values |
|---|
| 721 | hc = wc - zc #Water depths at centroids |
|---|
| 722 | |
|---|
| 723 | #Update |
|---|
| 724 | for k in range(domain.number_of_elements): |
|---|
| 725 | |
|---|
| 726 | if hc[k] < domain.minimum_allowed_height: |
|---|
| 727 | #Control stage |
|---|
| 728 | if hc[k] < domain.epsilon: |
|---|
| 729 | wc[k] = zc[k] # Contain 'lost mass' error |
|---|
| 730 | |
|---|
| 731 | #Control momentum |
|---|
| 732 | #xmomc[k] = ymomc[k] = 0.0 |
|---|
| 733 | xmomc[k] = 0.0 |
|---|
| 734 | |
|---|
| 735 | def protect_against_infinitesimal_and_negative_heights_c(domain): |
|---|
| 736 | """Protect against infinitesimal heights and associated high velocities |
|---|
| 737 | """ |
|---|
| 738 | |
|---|
| 739 | #Shortcuts |
|---|
| 740 | wc = domain.quantities['stage'].centroid_values |
|---|
| 741 | zc = domain.quantities['elevation'].centroid_values |
|---|
| 742 | xmomc = domain.quantities['xmomentum'].centroid_values |
|---|
| 743 | # ymomc = domain.quantities['ymomentum'].centroid_values |
|---|
| 744 | |
|---|
| 745 | from shallow_water_ext import protect |
|---|
| 746 | |
|---|
| 747 | protect(domain.minimum_allowed_height, domain.epsilon, |
|---|
| 748 | wc, zc, xmomc)#, ymomc) |
|---|
| 749 | |
|---|
| 750 | def h_limiter(domain): |
|---|
| 751 | """Limit slopes for each volume to eliminate artificial variance |
|---|
| 752 | introduced by e.g. second order extrapolator |
|---|
| 753 | |
|---|
| 754 | limit on h = w-z |
|---|
| 755 | |
|---|
| 756 | This limiter depends on two quantities (w,z) so it resides within |
|---|
| 757 | this module rather than within quantity.py |
|---|
| 758 | """ |
|---|
| 759 | |
|---|
| 760 | from Numeric import zeros, Float |
|---|
| 761 | |
|---|
| 762 | N = domain.number_of_elements |
|---|
| 763 | beta_h = domain.beta_h |
|---|
| 764 | |
|---|
| 765 | #Shortcuts |
|---|
| 766 | wc = domain.quantities['stage'].centroid_values |
|---|
| 767 | zc = domain.quantities['elevation'].centroid_values |
|---|
| 768 | hc = wc - zc |
|---|
| 769 | |
|---|
| 770 | wv = domain.quantities['stage'].vertex_values |
|---|
| 771 | zv = domain.quantities['elevation'].vertex_values |
|---|
| 772 | hv = wv-zv |
|---|
| 773 | |
|---|
| 774 | hvbar = zeros(hv.shape, Float) #h-limited values |
|---|
| 775 | |
|---|
| 776 | #Find min and max of this and neighbour's centroid values |
|---|
| 777 | hmax = zeros(hc.shape, Float) |
|---|
| 778 | hmin = zeros(hc.shape, Float) |
|---|
| 779 | |
|---|
| 780 | for k in range(N): |
|---|
| 781 | hmax[k] = hmin[k] = hc[k] |
|---|
| 782 | #for i in range(3): |
|---|
| 783 | for i in range(2): |
|---|
| 784 | n = domain.neighbours[k,i] |
|---|
| 785 | if n >= 0: |
|---|
| 786 | hn = hc[n] #Neighbour's centroid value |
|---|
| 787 | |
|---|
| 788 | hmin[k] = min(hmin[k], hn) |
|---|
| 789 | hmax[k] = max(hmax[k], hn) |
|---|
| 790 | |
|---|
| 791 | |
|---|
| 792 | #Diffences between centroids and maxima/minima |
|---|
| 793 | dhmax = hmax - hc |
|---|
| 794 | dhmin = hmin - hc |
|---|
| 795 | |
|---|
| 796 | #Deltas between vertex and centroid values |
|---|
| 797 | dh = zeros(hv.shape, Float) |
|---|
| 798 | #for i in range(3): |
|---|
| 799 | for i in range(2): |
|---|
| 800 | dh[:,i] = hv[:,i] - hc |
|---|
| 801 | |
|---|
| 802 | #Phi limiter |
|---|
| 803 | for k in range(N): |
|---|
| 804 | |
|---|
| 805 | #Find the gradient limiter (phi) across vertices |
|---|
| 806 | phi = 1.0 |
|---|
| 807 | #for i in range(3): |
|---|
| 808 | for i in range(2): |
|---|
| 809 | r = 1.0 |
|---|
| 810 | if (dh[k,i] > 0): r = dhmax[k]/dh[k,i] |
|---|
| 811 | if (dh[k,i] < 0): r = dhmin[k]/dh[k,i] |
|---|
| 812 | |
|---|
| 813 | phi = min( min(r*beta_h, 1), phi ) |
|---|
| 814 | |
|---|
| 815 | #Then update using phi limiter |
|---|
| 816 | #for i in range(3): |
|---|
| 817 | for i in range(2): |
|---|
| 818 | hvbar[k,i] = hc[k] + phi*dh[k,i] |
|---|
| 819 | |
|---|
| 820 | return hvbar |
|---|
| 821 | |
|---|
| 822 | |
|---|
| 823 | |
|---|
| 824 | def h_limiter_c(domain): |
|---|
| 825 | """Limit slopes for each volume to eliminate artificial variance |
|---|
| 826 | introduced by e.g. second order extrapolator |
|---|
| 827 | |
|---|
| 828 | limit on h = w-z |
|---|
| 829 | |
|---|
| 830 | This limiter depends on two quantities (w,z) so it resides within |
|---|
| 831 | this module rather than within quantity.py |
|---|
| 832 | |
|---|
| 833 | Wrapper for c-extension |
|---|
| 834 | """ |
|---|
| 835 | |
|---|
| 836 | from Numeric import zeros, Float |
|---|
| 837 | |
|---|
| 838 | N = domain.number_of_elements |
|---|
| 839 | beta_h = domain.beta_h |
|---|
| 840 | |
|---|
| 841 | #Shortcuts |
|---|
| 842 | wc = domain.quantities['stage'].centroid_values |
|---|
| 843 | zc = domain.quantities['elevation'].centroid_values |
|---|
| 844 | hc = wc - zc |
|---|
| 845 | |
|---|
| 846 | wv = domain.quantities['stage'].vertex_values |
|---|
| 847 | zv = domain.quantities['elevation'].vertex_values |
|---|
| 848 | hv = wv - zv |
|---|
| 849 | |
|---|
| 850 | #Call C-extension |
|---|
| 851 | from shallow_water_ext import h_limiter_sw as h_limiter |
|---|
| 852 | hvbar = h_limiter(domain, hc, hv) |
|---|
| 853 | |
|---|
| 854 | return hvbar |
|---|
| 855 | |
|---|
| 856 | def balance_deep_and_shallow(domain): |
|---|
| 857 | """Compute linear combination between stage as computed by |
|---|
| 858 | gradient-limiters limiting using w, and stage computed by |
|---|
| 859 | gradient-limiters limiting using h (h-limiter). |
|---|
| 860 | The former takes precedence when heights are large compared to the |
|---|
| 861 | bed slope while the latter takes precedence when heights are |
|---|
| 862 | relatively small. Anything in between is computed as a balanced |
|---|
| 863 | linear combination in order to avoid numerical disturbances which |
|---|
| 864 | would otherwise appear as a result of hard switching between |
|---|
| 865 | modes. |
|---|
| 866 | |
|---|
| 867 | The h-limiter is always applied irrespective of the order. |
|---|
| 868 | """ |
|---|
| 869 | |
|---|
| 870 | #Shortcuts |
|---|
| 871 | wc = domain.quantities['stage'].centroid_values |
|---|
| 872 | zc = domain.quantities['elevation'].centroid_values |
|---|
| 873 | hc = wc - zc |
|---|
| 874 | |
|---|
| 875 | wv = domain.quantities['stage'].vertex_values |
|---|
| 876 | zv = domain.quantities['elevation'].vertex_values |
|---|
| 877 | hv = wv-zv |
|---|
| 878 | |
|---|
| 879 | #Limit h |
|---|
| 880 | hvbar = h_limiter(domain) |
|---|
| 881 | |
|---|
| 882 | for k in range(domain.number_of_elements): |
|---|
| 883 | #Compute maximal variation in bed elevation |
|---|
| 884 | # This quantitiy is |
|---|
| 885 | # dz = max_i abs(z_i - z_c) |
|---|
| 886 | # and it is independent of dimension |
|---|
| 887 | # In the 1d case zc = (z0+z1)/2 |
|---|
| 888 | # In the 2d case zc = (z0+z1+z2)/3 |
|---|
| 889 | |
|---|
| 890 | dz = max(abs(zv[k,0]-zc[k]), |
|---|
| 891 | abs(zv[k,1]-zc[k]))#, |
|---|
| 892 | # abs(zv[k,2]-zc[k])) |
|---|
| 893 | |
|---|
| 894 | |
|---|
| 895 | hmin = min( hv[k,:] ) |
|---|
| 896 | |
|---|
| 897 | #Create alpha in [0,1], where alpha==0 means using the h-limited |
|---|
| 898 | #stage and alpha==1 means using the w-limited stage as |
|---|
| 899 | #computed by the gradient limiter (both 1st or 2nd order) |
|---|
| 900 | |
|---|
| 901 | #If hmin > dz/2 then alpha = 1 and the bed will have no effect |
|---|
| 902 | #If hmin < 0 then alpha = 0 reverting to constant height above bed. |
|---|
| 903 | |
|---|
| 904 | if dz > 0.0: |
|---|
| 905 | alpha = max( min( 2*hmin/dz, 1.0), 0.0 ) |
|---|
| 906 | else: |
|---|
| 907 | #Flat bed |
|---|
| 908 | alpha = 1.0 |
|---|
| 909 | |
|---|
| 910 | #Let |
|---|
| 911 | # |
|---|
| 912 | # wvi be the w-limited stage (wvi = zvi + hvi) |
|---|
| 913 | # wvi- be the h-limited state (wvi- = zvi + hvi-) |
|---|
| 914 | # |
|---|
| 915 | # |
|---|
| 916 | #where i=0,1,2 denotes the vertex ids |
|---|
| 917 | # |
|---|
| 918 | #Weighted balance between w-limited and h-limited stage is |
|---|
| 919 | # |
|---|
| 920 | # wvi := (1-alpha)*(zvi+hvi-) + alpha*(zvi+hvi) |
|---|
| 921 | # |
|---|
| 922 | #It follows that the updated wvi is |
|---|
| 923 | # wvi := zvi + (1-alpha)*hvi- + alpha*hvi |
|---|
| 924 | # |
|---|
| 925 | # Momentum is balanced between constant and limited |
|---|
| 926 | |
|---|
| 927 | |
|---|
| 928 | #for i in range(3): |
|---|
| 929 | # wv[k,i] = zv[k,i] + hvbar[k,i] |
|---|
| 930 | |
|---|
| 931 | #return |
|---|
| 932 | |
|---|
| 933 | if alpha < 1: |
|---|
| 934 | |
|---|
| 935 | #for i in range(3): |
|---|
| 936 | for i in range(2): |
|---|
| 937 | wv[k,i] = zv[k,i] + (1-alpha)*hvbar[k,i] + alpha*hv[k,i] |
|---|
| 938 | |
|---|
| 939 | #Momentums at centroids |
|---|
| 940 | xmomc = domain.quantities['xmomentum'].centroid_values |
|---|
| 941 | # ymomc = domain.quantities['ymomentum'].centroid_values |
|---|
| 942 | |
|---|
| 943 | #Momentums at vertices |
|---|
| 944 | xmomv = domain.quantities['xmomentum'].vertex_values |
|---|
| 945 | # ymomv = domain.quantities['ymomentum'].vertex_values |
|---|
| 946 | |
|---|
| 947 | # Update momentum as a linear combination of |
|---|
| 948 | # xmomc and ymomc (shallow) and momentum |
|---|
| 949 | # from extrapolator xmomv and ymomv (deep). |
|---|
| 950 | xmomv[k,:] = (1-alpha)*xmomc[k] + alpha*xmomv[k,:] |
|---|
| 951 | # ymomv[k,:] = (1-alpha)*ymomc[k] + alpha*ymomv[k,:] |
|---|
| 952 | |
|---|
| 953 | |
|---|
| 954 | def balance_deep_and_shallow_c(domain): |
|---|
| 955 | """Wrapper for C implementation |
|---|
| 956 | """ |
|---|
| 957 | |
|---|
| 958 | #Shortcuts |
|---|
| 959 | wc = domain.quantities['stage'].centroid_values |
|---|
| 960 | zc = domain.quantities['elevation'].centroid_values |
|---|
| 961 | hc = wc - zc |
|---|
| 962 | |
|---|
| 963 | wv = domain.quantities['stage'].vertex_values |
|---|
| 964 | zv = domain.quantities['elevation'].vertex_values |
|---|
| 965 | hv = wv - zv |
|---|
| 966 | |
|---|
| 967 | #Momentums at centroids |
|---|
| 968 | xmomc = domain.quantities['xmomentum'].centroid_values |
|---|
| 969 | # ymomc = domain.quantities['ymomentum'].centroid_values |
|---|
| 970 | |
|---|
| 971 | #Momentums at vertices |
|---|
| 972 | xmomv = domain.quantities['xmomentum'].vertex_values |
|---|
| 973 | # ymomv = domain.quantities['ymomentum'].vertex_values |
|---|
| 974 | |
|---|
| 975 | #Limit h |
|---|
| 976 | hvbar = h_limiter(domain) |
|---|
| 977 | |
|---|
| 978 | #This is how one would make a first order h_limited value |
|---|
| 979 | #as in the old balancer (pre 17 Feb 2005): |
|---|
| 980 | #from Numeric import zeros, Float |
|---|
| 981 | #hvbar = zeros( (len(hc), 3), Float) |
|---|
| 982 | #for i in range(3): |
|---|
| 983 | # hvbar[:,i] = hc[:] |
|---|
| 984 | |
|---|
| 985 | from shallow_water_ext import balance_deep_and_shallow |
|---|
| 986 | balance_deep_and_shallow(wc, zc, hc, wv, zv, hv, hvbar, |
|---|
| 987 | # xmomc, ymomc, xmomv, ymomv) |
|---|
| 988 | xmomc, xmomv) |
|---|
| 989 | |
|---|
| 990 | |
|---|
| 991 | ############################################### |
|---|
| 992 | #Boundaries - specific to the shallow water wave equation |
|---|
| 993 | class Reflective_boundary(Boundary): |
|---|
| 994 | """Reflective boundary returns same conserved quantities as |
|---|
| 995 | those present in its neighbour volume but reflected. |
|---|
| 996 | |
|---|
| 997 | This class is specific to the shallow water equation as it |
|---|
| 998 | works with the momentum quantities assumed to be the second |
|---|
| 999 | and third conserved quantities. |
|---|
| 1000 | """ |
|---|
| 1001 | |
|---|
| 1002 | def __init__(self, domain = None): |
|---|
| 1003 | Boundary.__init__(self) |
|---|
| 1004 | |
|---|
| 1005 | if domain is None: |
|---|
| 1006 | msg = 'Domain must be specified for reflective boundary' |
|---|
| 1007 | raise msg |
|---|
| 1008 | |
|---|
| 1009 | #Handy shorthands |
|---|
| 1010 | #self.stage = domain.quantities['stage'].edge_values |
|---|
| 1011 | #self.xmom = domain.quantities['xmomentum'].edge_values |
|---|
| 1012 | #self.ymom = domain.quantities['ymomentum'].edge_values |
|---|
| 1013 | #self.normals = domain.normals |
|---|
| 1014 | self.stage = domain.quantities['stage'].vertex_values |
|---|
| 1015 | self.xmom = domain.quantities['xmomentum'].vertex_values |
|---|
| 1016 | |
|---|
| 1017 | from Numeric import zeros, Float |
|---|
| 1018 | #self.conserved_quantities = zeros(3, Float) |
|---|
| 1019 | self.conserved_quantities = zeros(2, Float) |
|---|
| 1020 | |
|---|
| 1021 | def __repr__(self): |
|---|
| 1022 | return 'Reflective_boundary' |
|---|
| 1023 | |
|---|
| 1024 | |
|---|
| 1025 | def evaluate(self, vol_id, edge_id): |
|---|
| 1026 | """Reflective boundaries reverses the outward momentum |
|---|
| 1027 | of the volume they serve. |
|---|
| 1028 | """ |
|---|
| 1029 | |
|---|
| 1030 | q = self.conserved_quantities |
|---|
| 1031 | q[0] = self.stage[vol_id, edge_id] |
|---|
| 1032 | q[1] = self.xmom[vol_id, edge_id] |
|---|
| 1033 | #q[2] = self.ymom[vol_id, edge_id] |
|---|
| 1034 | |
|---|
| 1035 | #normal = self.normals[vol_id, 2*edge_id:2*edge_id+2] |
|---|
| 1036 | #normal = self.normals[vol_id, 2*edge_id:2*edge_id+1] |
|---|
| 1037 | |
|---|
| 1038 | |
|---|
| 1039 | #r = rotate(q, normal, direction = 1) |
|---|
| 1040 | #r[1] = -r[1] |
|---|
| 1041 | #q = rotate(r, normal, direction = -1) |
|---|
| 1042 | r = q |
|---|
| 1043 | r[1] = -q[1] |
|---|
| 1044 | q = r |
|---|
| 1045 | #For start interval there is no outward momentum so do not need to |
|---|
| 1046 | #reverse direction in this case |
|---|
| 1047 | |
|---|
| 1048 | return q |
|---|
| 1049 | |
|---|
| 1050 | |
|---|
| 1051 | ######################### |
|---|
| 1052 | #Standard forcing terms: |
|---|
| 1053 | # |
|---|
| 1054 | def gravity(domain): |
|---|
| 1055 | """Apply gravitational pull in the presence of bed slope |
|---|
| 1056 | """ |
|---|
| 1057 | |
|---|
| 1058 | from util import gradient |
|---|
| 1059 | from Numeric import zeros, Float, array, sum |
|---|
| 1060 | |
|---|
| 1061 | xmom = domain.quantities['xmomentum'].explicit_update |
|---|
| 1062 | # ymom = domain.quantities['ymomentum'].explicit_update |
|---|
| 1063 | |
|---|
| 1064 | Stage = domain.quantities['stage'] |
|---|
| 1065 | Elevation = domain.quantities['elevation'] |
|---|
| 1066 | #h = Stage.edge_values - Elevation.edge_values |
|---|
| 1067 | h = Stage.vertex_values - Elevation.vertex_values |
|---|
| 1068 | b = Elevation.vertex_values |
|---|
| 1069 | w = Stage.vertex_values |
|---|
| 1070 | |
|---|
| 1071 | x = domain.get_vertex_coordinates() |
|---|
| 1072 | g = domain.g |
|---|
| 1073 | |
|---|
| 1074 | for k in range(domain.number_of_elements): |
|---|
| 1075 | # avg_h = sum( h[k,:] )/3 |
|---|
| 1076 | avg_h = sum( h[k,:] )/2 |
|---|
| 1077 | |
|---|
| 1078 | #Compute bed slope |
|---|
| 1079 | #x0, y0, x1, y1, x2, y2 = x[k,:] |
|---|
| 1080 | x0, x1 = x[k,:] |
|---|
| 1081 | #z0, z1, z2 = v[k,:] |
|---|
| 1082 | b0, b1 = b[k,:] |
|---|
| 1083 | |
|---|
| 1084 | w0, w1 = w[k,:] |
|---|
| 1085 | wx = gradient(x0, x1, w0, w1) |
|---|
| 1086 | |
|---|
| 1087 | #zx, zy = gradient(x0, y0, x1, y1, x2, y2, z0, z1, z2) |
|---|
| 1088 | bx = gradient(x0, x1, b0, b1) |
|---|
| 1089 | |
|---|
| 1090 | #Update momentum |
|---|
| 1091 | if domain.fluxfunc == "unsplit": |
|---|
| 1092 | xmom[k] += -g*bx*avg_h |
|---|
| 1093 | elif domain.fluxfunc == "split": |
|---|
| 1094 | xmom[k] += -g*wx*avg_h |
|---|
| 1095 | # ymom[k] += -g*zy*avg_h |
|---|
| 1096 | |
|---|
| 1097 | |
|---|
| 1098 | def graity_c(domain): |
|---|
| 1099 | """Wrapper calling C version |
|---|
| 1100 | """ |
|---|
| 1101 | |
|---|
| 1102 | xmom = domain.quantities['xmomentum'].explicit_update |
|---|
| 1103 | # ymom = domain.quantities['ymomentum'].explicit_update |
|---|
| 1104 | |
|---|
| 1105 | Stage = domain.quantities['stage'] |
|---|
| 1106 | Elevation = domain.quantities['elevation'] |
|---|
| 1107 | h = Stage.edge_values - Elevation.edge_values |
|---|
| 1108 | v = Elevation.vertex_values |
|---|
| 1109 | |
|---|
| 1110 | x = domain.get_vertex_coordinates() |
|---|
| 1111 | g = domain.g |
|---|
| 1112 | |
|---|
| 1113 | |
|---|
| 1114 | from shallow_water_ext import gravity |
|---|
| 1115 | gravity(g, h, v, x, xmom, ymom) |
|---|
| 1116 | |
|---|
| 1117 | |
|---|
| 1118 | def manning_friction(domain): |
|---|
| 1119 | """Apply (Manning) friction to water momentum |
|---|
| 1120 | """ |
|---|
| 1121 | |
|---|
| 1122 | from math import sqrt |
|---|
| 1123 | |
|---|
| 1124 | w = domain.quantities['stage'].centroid_values |
|---|
| 1125 | z = domain.quantities['elevation'].centroid_values |
|---|
| 1126 | h = w-z |
|---|
| 1127 | |
|---|
| 1128 | uh = domain.quantities['xmomentum'].centroid_values |
|---|
| 1129 | #vh = domain.quantities['ymomentum'].centroid_values |
|---|
| 1130 | eta = domain.quantities['friction'].centroid_values |
|---|
| 1131 | |
|---|
| 1132 | xmom_update = domain.quantities['xmomentum'].semi_implicit_update |
|---|
| 1133 | #ymom_update = domain.quantities['ymomentum'].semi_implicit_update |
|---|
| 1134 | |
|---|
| 1135 | N = domain.number_of_elements |
|---|
| 1136 | eps = domain.minimum_allowed_height |
|---|
| 1137 | g = domain.g |
|---|
| 1138 | |
|---|
| 1139 | for k in range(N): |
|---|
| 1140 | if eta[k] >= eps: |
|---|
| 1141 | if h[k] >= eps: |
|---|
| 1142 | #S = -g * eta[k]**2 * sqrt((uh[k]**2 + vh[k]**2)) |
|---|
| 1143 | S = -g * eta[k]**2 * uh[k] |
|---|
| 1144 | S /= h[k]**(7.0/3) |
|---|
| 1145 | |
|---|
| 1146 | #Update momentum |
|---|
| 1147 | xmom_update[k] += S*uh[k] |
|---|
| 1148 | #ymom_update[k] += S*vh[k] |
|---|
| 1149 | |
|---|
| 1150 | |
|---|
| 1151 | def manning_friction_c(domain): |
|---|
| 1152 | """Wrapper for c version |
|---|
| 1153 | """ |
|---|
| 1154 | |
|---|
| 1155 | xmom = domain.quantities['xmomentum'] |
|---|
| 1156 | # ymom = domain.quantities['ymomentum'] |
|---|
| 1157 | |
|---|
| 1158 | w = domain.quantities['stage'].centroid_values |
|---|
| 1159 | z = domain.quantities['elevation'].centroid_values |
|---|
| 1160 | |
|---|
| 1161 | uh = xmom.centroid_values |
|---|
| 1162 | # vh = ymom.centroid_values |
|---|
| 1163 | eta = domain.quantities['friction'].centroid_values |
|---|
| 1164 | |
|---|
| 1165 | xmom_update = xmom.semi_implicit_update |
|---|
| 1166 | # ymom_update = ymom.semi_implicit_update |
|---|
| 1167 | |
|---|
| 1168 | N = domain.number_of_elements |
|---|
| 1169 | eps = domain.minimum_allowed_height |
|---|
| 1170 | g = domain.g |
|---|
| 1171 | |
|---|
| 1172 | from shallow_water_ext import manning_friction |
|---|
| 1173 | # manning_friction(g, eps, w, z, uh, vh, eta, xmom_update, ymom_update) |
|---|
| 1174 | manning_friction(g, eps, w, z, uh, eta, xmom_update) |
|---|
| 1175 | |
|---|
| 1176 | def linear_friction(domain): |
|---|
| 1177 | """Apply linear friction to water momentum |
|---|
| 1178 | |
|---|
| 1179 | Assumes quantity: 'linear_friction' to be present |
|---|
| 1180 | """ |
|---|
| 1181 | |
|---|
| 1182 | from math import sqrt |
|---|
| 1183 | |
|---|
| 1184 | w = domain.quantities['stage'].centroid_values |
|---|
| 1185 | z = domain.quantities['elevation'].centroid_values |
|---|
| 1186 | h = w-z |
|---|
| 1187 | |
|---|
| 1188 | uh = domain.quantities['xmomentum'].centroid_values |
|---|
| 1189 | # vh = domain.quantities['ymomentum'].centroid_values |
|---|
| 1190 | tau = domain.quantities['linear_friction'].centroid_values |
|---|
| 1191 | |
|---|
| 1192 | xmom_update = domain.quantities['xmomentum'].semi_implicit_update |
|---|
| 1193 | # ymom_update = domain.quantities['ymomentum'].semi_implicit_update |
|---|
| 1194 | |
|---|
| 1195 | N = domain.number_of_elements |
|---|
| 1196 | eps = domain.minimum_allowed_height |
|---|
| 1197 | g = domain.g #Not necessary? Why was this added? |
|---|
| 1198 | |
|---|
| 1199 | for k in range(N): |
|---|
| 1200 | if tau[k] >= eps: |
|---|
| 1201 | if h[k] >= eps: |
|---|
| 1202 | S = -tau[k]/h[k] |
|---|
| 1203 | |
|---|
| 1204 | #Update momentum |
|---|
| 1205 | xmom_update[k] += S*uh[k] |
|---|
| 1206 | # ymom_update[k] += S*vh[k] |
|---|
| 1207 | |
|---|
| 1208 | |
|---|
| 1209 | |
|---|
| 1210 | def check_forcefield(f): |
|---|
| 1211 | """Check that f is either |
|---|
| 1212 | 1: a callable object f(t,x,y), where x and y are vectors |
|---|
| 1213 | and that it returns an array or a list of same length |
|---|
| 1214 | as x and y |
|---|
| 1215 | 2: a scalar |
|---|
| 1216 | """ |
|---|
| 1217 | |
|---|
| 1218 | from Numeric import ones, Float, array |
|---|
| 1219 | |
|---|
| 1220 | |
|---|
| 1221 | if callable(f): |
|---|
| 1222 | #N = 3 |
|---|
| 1223 | N = 2 |
|---|
| 1224 | #x = ones(3, Float) |
|---|
| 1225 | #y = ones(3, Float) |
|---|
| 1226 | x = ones(2, Float) |
|---|
| 1227 | #y = ones(2, Float) |
|---|
| 1228 | |
|---|
| 1229 | try: |
|---|
| 1230 | #q = f(1.0, x=x, y=y) |
|---|
| 1231 | q = f(1.0, x=x) |
|---|
| 1232 | except Exception, e: |
|---|
| 1233 | msg = 'Function %s could not be executed:\n%s' %(f, e) |
|---|
| 1234 | #FIXME: Reconsider this semantics |
|---|
| 1235 | raise msg |
|---|
| 1236 | |
|---|
| 1237 | try: |
|---|
| 1238 | q = array(q).astype(Float) |
|---|
| 1239 | except: |
|---|
| 1240 | msg = 'Return value from vector function %s could ' %f |
|---|
| 1241 | msg += 'not be converted into a Numeric array of floats.\n' |
|---|
| 1242 | msg += 'Specified function should return either list or array.' |
|---|
| 1243 | raise msg |
|---|
| 1244 | |
|---|
| 1245 | #Is this really what we want? |
|---|
| 1246 | msg = 'Return vector from function %s ' %f |
|---|
| 1247 | msg += 'must have same lenght as input vectors' |
|---|
| 1248 | assert len(q) == N, msg |
|---|
| 1249 | |
|---|
| 1250 | else: |
|---|
| 1251 | try: |
|---|
| 1252 | f = float(f) |
|---|
| 1253 | except: |
|---|
| 1254 | msg = 'Force field %s must be either a scalar' %f |
|---|
| 1255 | msg += ' or a vector function' |
|---|
| 1256 | raise msg |
|---|
| 1257 | return f |
|---|
| 1258 | |
|---|
| 1259 | class Wind_stress: |
|---|
| 1260 | """Apply wind stress to water momentum in terms of |
|---|
| 1261 | wind speed [m/s] and wind direction [degrees] |
|---|
| 1262 | """ |
|---|
| 1263 | |
|---|
| 1264 | def __init__(self, *args, **kwargs): |
|---|
| 1265 | """Initialise windfield from wind speed s [m/s] |
|---|
| 1266 | and wind direction phi [degrees] |
|---|
| 1267 | |
|---|
| 1268 | Inputs v and phi can be either scalars or Python functions, e.g. |
|---|
| 1269 | |
|---|
| 1270 | W = Wind_stress(10, 178) |
|---|
| 1271 | |
|---|
| 1272 | #FIXME - 'normal' degrees are assumed for now, i.e. the |
|---|
| 1273 | vector (1,0) has zero degrees. |
|---|
| 1274 | We may need to convert from 'compass' degrees later on and also |
|---|
| 1275 | map from True north to grid north. |
|---|
| 1276 | |
|---|
| 1277 | Arguments can also be Python functions of t,x,y as in |
|---|
| 1278 | |
|---|
| 1279 | def speed(t,x,y): |
|---|
| 1280 | ... |
|---|
| 1281 | return s |
|---|
| 1282 | |
|---|
| 1283 | def angle(t,x,y): |
|---|
| 1284 | ... |
|---|
| 1285 | return phi |
|---|
| 1286 | |
|---|
| 1287 | where x and y are vectors. |
|---|
| 1288 | |
|---|
| 1289 | and then pass the functions in |
|---|
| 1290 | |
|---|
| 1291 | W = Wind_stress(speed, angle) |
|---|
| 1292 | |
|---|
| 1293 | The instantiated object W can be appended to the list of |
|---|
| 1294 | forcing_terms as in |
|---|
| 1295 | |
|---|
| 1296 | Alternatively, one vector valued function for (speed, angle) |
|---|
| 1297 | can be applied, providing both quantities simultaneously. |
|---|
| 1298 | As in |
|---|
| 1299 | W = Wind_stress(F), where returns (speed, angle) for each t. |
|---|
| 1300 | |
|---|
| 1301 | domain.forcing_terms.append(W) |
|---|
| 1302 | """ |
|---|
| 1303 | |
|---|
| 1304 | from config import rho_a, rho_w, eta_w |
|---|
| 1305 | from Numeric import array, Float |
|---|
| 1306 | |
|---|
| 1307 | if len(args) == 2: |
|---|
| 1308 | s = args[0] |
|---|
| 1309 | phi = args[1] |
|---|
| 1310 | elif len(args) == 1: |
|---|
| 1311 | #Assume vector function returning (s, phi)(t,x,y) |
|---|
| 1312 | vector_function = args[0] |
|---|
| 1313 | #s = lambda t,x,y: vector_function(t,x=x,y=y)[0] |
|---|
| 1314 | #phi = lambda t,x,y: vector_function(t,x=x,y=y)[1] |
|---|
| 1315 | s = lambda t,x: vector_function(t,x=x)[0] |
|---|
| 1316 | phi = lambda t,x: vector_function(t,x=x)[1] |
|---|
| 1317 | else: |
|---|
| 1318 | #Assume info is in 2 keyword arguments |
|---|
| 1319 | |
|---|
| 1320 | if len(kwargs) == 2: |
|---|
| 1321 | s = kwargs['s'] |
|---|
| 1322 | phi = kwargs['phi'] |
|---|
| 1323 | else: |
|---|
| 1324 | raise 'Assumes two keyword arguments: s=..., phi=....' |
|---|
| 1325 | |
|---|
| 1326 | print 'phi', phi |
|---|
| 1327 | self.speed = check_forcefield(s) |
|---|
| 1328 | self.phi = check_forcefield(phi) |
|---|
| 1329 | |
|---|
| 1330 | self.const = eta_w*rho_a/rho_w |
|---|
| 1331 | |
|---|
| 1332 | |
|---|
| 1333 | def __call__(self, domain): |
|---|
| 1334 | """Evaluate windfield based on values found in domain |
|---|
| 1335 | """ |
|---|
| 1336 | |
|---|
| 1337 | from math import pi, cos, sin, sqrt |
|---|
| 1338 | from Numeric import Float, ones, ArrayType |
|---|
| 1339 | |
|---|
| 1340 | xmom_update = domain.quantities['xmomentum'].explicit_update |
|---|
| 1341 | #ymom_update = domain.quantities['ymomentum'].explicit_update |
|---|
| 1342 | |
|---|
| 1343 | N = domain.number_of_elements |
|---|
| 1344 | t = domain.time |
|---|
| 1345 | |
|---|
| 1346 | if callable(self.speed): |
|---|
| 1347 | xc = domain.get_centroid_coordinates() |
|---|
| 1348 | #s_vec = self.speed(t, xc[:,0], xc[:,1]) |
|---|
| 1349 | s_vec = self.speed(t, xc) |
|---|
| 1350 | else: |
|---|
| 1351 | #Assume s is a scalar |
|---|
| 1352 | |
|---|
| 1353 | try: |
|---|
| 1354 | s_vec = self.speed * ones(N, Float) |
|---|
| 1355 | except: |
|---|
| 1356 | msg = 'Speed must be either callable or a scalar: %s' %self.s |
|---|
| 1357 | raise msg |
|---|
| 1358 | |
|---|
| 1359 | |
|---|
| 1360 | if callable(self.phi): |
|---|
| 1361 | xc = domain.get_centroid_coordinates() |
|---|
| 1362 | #phi_vec = self.phi(t, xc[:,0], xc[:,1]) |
|---|
| 1363 | phi_vec = self.phi(t, xc) |
|---|
| 1364 | else: |
|---|
| 1365 | #Assume phi is a scalar |
|---|
| 1366 | |
|---|
| 1367 | try: |
|---|
| 1368 | phi_vec = self.phi * ones(N, Float) |
|---|
| 1369 | except: |
|---|
| 1370 | msg = 'Angle must be either callable or a scalar: %s' %self.phi |
|---|
| 1371 | raise msg |
|---|
| 1372 | |
|---|
| 1373 | #assign_windfield_values(xmom_update, ymom_update, |
|---|
| 1374 | # s_vec, phi_vec, self.const) |
|---|
| 1375 | assign_windfield_values(xmom_update, s_vec, phi_vec, self.const) |
|---|
| 1376 | |
|---|
| 1377 | |
|---|
| 1378 | #def assign_windfield_values(xmom_update, ymom_update, |
|---|
| 1379 | # s_vec, phi_vec, const): |
|---|
| 1380 | def assign_windfield_values(xmom_update, s_vec, phi_vec, const): |
|---|
| 1381 | """Python version of assigning wind field to update vectors. |
|---|
| 1382 | A c version also exists (for speed) |
|---|
| 1383 | """ |
|---|
| 1384 | from math import pi, cos, sin, sqrt |
|---|
| 1385 | |
|---|
| 1386 | N = len(s_vec) |
|---|
| 1387 | for k in range(N): |
|---|
| 1388 | s = s_vec[k] |
|---|
| 1389 | phi = phi_vec[k] |
|---|
| 1390 | |
|---|
| 1391 | #Convert to radians |
|---|
| 1392 | phi = phi*pi/180 |
|---|
| 1393 | |
|---|
| 1394 | #Compute velocity vector (u, v) |
|---|
| 1395 | u = s*cos(phi) |
|---|
| 1396 | v = s*sin(phi) |
|---|
| 1397 | |
|---|
| 1398 | #Compute wind stress |
|---|
| 1399 | #S = const * sqrt(u**2 + v**2) |
|---|
| 1400 | S = const * u |
|---|
| 1401 | xmom_update[k] += S*u |
|---|
| 1402 | #ymom_update[k] += S*v |
|---|
