├── .gitignore ├── LICENSE ├── README.md ├── model.py └── run.py /.gitignore: -------------------------------------------------------------------------------- 1 | *.pyc 2 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # QGModel 2 | This is Python code that solves the nonlinear QG equations for PV anomalies on a doubly-periodic domain. It consists of two parts: 3 | 4 | 1. a fully-dealiased pseudo-spectral code that steps forward the PV conservation equations and 5 | 2. implementations of specific model types that provide an inversion relation. 6 | 7 | At the moment, seven model types have been implemented: 8 | 9 | 1. two-dimensional dynamics (`TwoDim`), 10 | 2. surface QG dynamics (`Surface`), 11 | 3. multi-layer dynamics (`Layered`), 12 | 4. Eady dynamics (`Eady`), 13 | 5. floating Eady dynamics (`FloatingEady`), 14 | 6. two-Eady dynamics (`TwoEady`), 15 | 7. two-Eady dynamics with buoyancy jump (`TwoEadyJump`). 16 | 17 | See `run.py` for an example of how a model is initialized and run. 18 | 19 | This model makes use of [PyFFTW](https://pypi.python.org/pypi/pyFFTW), a Python wrapper of [FFTW](http://www.fftw.org/). Follow the [installation instructions](https://github.com/hgomersall/pyFFTW) for PyFFTW if you do not have it installed already. PyFFTW will speed up the code and allows multi-threading, but the model will also run with standard numpy FFT routines. 20 | -------------------------------------------------------------------------------- /model.py: -------------------------------------------------------------------------------- 1 | # Copyright (C) 2013,2014,2015 Joern Callies 2 | # 3 | # This file is part of QGModel. 4 | # 5 | # QGModel is free software: you can redistribute it and/or modify 6 | # it under the terms of the GNU General Public License as published by 7 | # the Free Software Foundation, either version 3 of the License, or 8 | # (at your option) any later version. 9 | # 10 | # QGModel is distributed in the hope that it will be useful, 11 | # but WITHOUT ANY WARRANTY; without even the implied warranty of 12 | # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 13 | # GNU General Public License for more details. 14 | # 15 | # You should have received a copy of the GNU General Public License 16 | # along with QGModel. If not, see . 17 | 18 | 19 | import sys 20 | import os 21 | import pickle 22 | 23 | import numpy as np 24 | import matplotlib.pyplot as plt 25 | import matplotlib.colors as mcl 26 | 27 | 28 | # Define blue colormap. 29 | cdict = { 30 | 'red': ((0.0, 0.0, 0.0), (0.5, 0.216, 0.216), (1.0, 1.0, 1.0)), 31 | 'green': ((0.0, 0.0, 0.0), (0.5, 0.494, 0.494), (1.0, 1.0, 1.0)), 32 | 'blue': ((0.0, 0.0, 0.0), (0.5, 0.722, 0.722), (1.0, 1.0, 1.0))} 33 | cm_blues = mcl.LinearSegmentedColormap('cm_blues', cdict, 256) 34 | 35 | 36 | class Model: 37 | 38 | """ 39 | General QG model 40 | 41 | This is the skeleton of a QG model that consists of a number of conserved 42 | quantities that are advected horizontally. Implementations of this model 43 | need to specify the number of conserved quantities (nz) and supply an 44 | inversion relation that yields the streamfuncion given the conserved quan- 45 | tities. The model geometry is doubly periodic in the perturbations; mean 46 | flow and gradients in the conserved quantities can be prescribed. 47 | """ 48 | 49 | def __init__(self, nz): 50 | self.nz = nz # number of levels 51 | self.u = np.zeros(nz) # mean zonal velocities 52 | self.v = np.zeros(nz) # mean meridional velocities 53 | self.qx = np.zeros(nz) # mean zonal PV gradient 54 | self.qy = np.zeros(nz) # mean meridional PV gradient 55 | 56 | def linstab(self, k, l): 57 | """ 58 | Perform linear stability analysis for wavenumbers k and l. 59 | 60 | Returns complex eigen-frequencies omega of the eigenvalue problem 61 | [k U + l V + (k Qy - l Qx) L^-1] q = omega q. 62 | The phase speeds are Re omega/k and the growth rates Im omega. 63 | """ 64 | # Set up inversion matrix with specified wavenumbers. 65 | L = self.invmatrix(k[np.newaxis,:,np.newaxis], 66 | l[:,np.newaxis,np.newaxis]) 67 | # Allow proper broadcasting over matrices. 68 | kk = k[np.newaxis,:,np.newaxis,np.newaxis] 69 | ll = l[:,np.newaxis,np.newaxis,np.newaxis] 70 | # Compute (k Gy - l Gx) L^-1. 71 | GL = np.einsum('...ij,...jk->...ik', kk*np.diag(self.qy) 72 | - ll*np.diag(self.qx), np.linalg.inv(L)) 73 | # Solve eigenvalue problem. 74 | w, v = np.linalg.eig(kk*np.diag(self.u) + ll*np.diag(self.v) + GL) 75 | # Sort eigenvalues. 76 | w.sort() 77 | return w 78 | 79 | def initnum(self, a, n, dt): 80 | """Initialize numerics.""" 81 | self.a = a # domain size 82 | self.n = n # number of Fourier modes per direction 83 | self.dt = dt # time step 84 | self.diffexp = 2 # exponent of diffusion operator 85 | self.hypodiff = 0. # hypodiffusion coefficient 86 | self.threads = 1 # number of threads for FFT 87 | self.clock = 0. # initial simulation time 88 | # Set up grid. 89 | self.grid() 90 | # Set up inversion matrix. 91 | self.L = self.invmatrix(self.k, self.l) 92 | 93 | def grid(self): 94 | """Set up spectral and physical grid.""" 95 | # Set up spectral grid. 96 | k = abs(np.fft.fftfreq(self.n, d=self.a/(2*np.pi*self.n))[:self.n/2+1]) 97 | l = np.fft.fftfreq(self.n, d=self.a/(2*np.pi*self.n)) 98 | self.k = k[np.newaxis,:,np.newaxis] 99 | self.l = l[:,np.newaxis,np.newaxis] 100 | # Set up physical grid. 101 | x = np.arange(self.n) * self.a / self.n 102 | y = np.arange(self.n) * self.a / self.n 103 | self.x = x[np.newaxis,:,np.newaxis] 104 | self.y = y[:,np.newaxis,np.newaxis] 105 | 106 | def initq(self, qp): 107 | """Transform qp to spectral space and initialize q.""" 108 | self.q = self.rfft2(qp, axes=(0, 1)) 109 | self.q[0,0,:] = 0. # ensuring zero mean 110 | 111 | def timestep(self): 112 | """Perform time step.""" 113 | self.advection() 114 | self.diffusion() 115 | self.clock += self.dt 116 | 117 | def advection(self): 118 | """Perform RK4 step for advective terms (linear and nonlinear).""" 119 | q1 = self.advrhs(self.q) 120 | q2 = self.advrhs(self.q + self.dt*q1/2) 121 | q3 = self.advrhs(self.q + self.dt*q2/2) 122 | q4 = self.advrhs(self.q + self.dt*q3) 123 | self.q += self.dt*(q1 + 2*q2 + 2*q3 + q4)/6 124 | 125 | def diffusion(self): 126 | """Perform implicit (hyper- and hypo-) diffusion step.""" 127 | k2 = self.k**2 + self.l**2 128 | k2[0,0,:] = 1. # preventing div. by zero for wavenumber 0 129 | self.q *= np.exp(-self.nu * k2**(self.diffexp/2.) * self.dt) 130 | if self.hypodiff > 0: 131 | self.q *= np.exp(-self.hypodiff / k2 * self.dt) 132 | 133 | def advrhs(self, q): 134 | """ 135 | Calculate advective terms on RHS of PV equation. 136 | 137 | Calculate mean-eddy and eddy-eddy advection terms: 138 | u q'_x + v q'_y + u' q_x + v' q_y + J(p', q') 139 | """ 140 | # Perform inversion. 141 | p = np.linalg.solve(self.L, q) 142 | # Calculate RHS. 143 | rhs = \ 144 | - 1j * (self.k*self.u + self.l*self.v) * q \ 145 | - 1j * (self.k*self.qy - self.l*self.qx) * p \ 146 | - self.jacobian(p, q) 147 | return rhs 148 | 149 | def jacobian(self, A, B): 150 | """ 151 | Calculate Jacobian A_x B_y - A_y B_x. 152 | 153 | Transform Ax, Ay, Bx, By to physical space, perform multi- 154 | plication and subtraction, and transform back to spectral space. 155 | To avoid aliasing, apply 3/2 padding. 156 | """ 157 | Axp = self.ifft_pad(1j * self.k * A) 158 | Ayp = self.ifft_pad(1j * self.l * A) 159 | Bxp = self.ifft_pad(1j * self.k * B) 160 | Byp = self.ifft_pad(1j * self.l * B) 161 | return self.fft_truncate(Axp * Byp - Ayp * Bxp) 162 | 163 | def fft_truncate(self, up): 164 | """Perform forward FFT on physical field up and truncate (3/2 rule).""" 165 | us = self.rfft2(up, axes=(0, 1)) 166 | u = np.zeros((self.n, self.n/2 + 1, self.nz), dtype=complex) 167 | u[: self.n/2, :, :] = us[: self.n/2, : self.n/2 + 1, :] 168 | u[self.n/2 :, :, :] = us[self.n : 3*self.n/2, : self.n/2 + 1, :] 169 | return u/2.25 # accounting for normalization 170 | 171 | def ifft_pad(self, u): 172 | """Pad spectral field u (3/2 rule) and perform inverse FFT.""" 173 | us = np.zeros((3*self.n/2, 3*self.n/4 + 1, self.nz), dtype=complex) 174 | us[: self.n/2, : self.n/2 + 1, :] = u[: self.n/2, :, :] 175 | us[self.n : 3*self.n/2, : self.n/2 + 1, :] = u[self.n/2 :, :, :] 176 | return self.irfft2(2.25*us, axes=(0, 1)) 177 | 178 | def rfft2(self, u, axes=(-2,-1)): 179 | """Real 2D FFT: use FFTW if available, otherwise numpy's FFT.""" 180 | try: 181 | import pyfftw as fftw 182 | us = fftw.interfaces.numpy_fft.rfft2(u, axes=axes, threads=self.threads) 183 | except ImportError: 184 | us = np.fft.rfft2(u, axes=axes) 185 | return us 186 | 187 | def irfft2(self, us, axes=(-2,-1)): 188 | """Real 2D IFFT: use FFTW if available, otherwise numpy's FFT.""" 189 | try: 190 | import pyfftw as fftw 191 | u = fftw.interfaces.numpy_fft.irfft2(us, axes=axes, threads=self.threads) 192 | except ImportError: 193 | u = np.fft.irfft2(us, axes=axes) 194 | return u 195 | 196 | def doubleres(self): 197 | """Double the resolution, interpolate fields.""" 198 | self.n *= 2 199 | # Pad spectral field. 200 | qs = np.zeros((self.n, self.n/2 + 1, self.nz), dtype=complex) 201 | qs[: self.n/4, : self.n/4 + 1, :] = self.q[: self.n/4, :, :] 202 | qs[3*self.n/4 : self.n, : self.n/4 + 1, :] = self.q[self.n/4 :, :, :] 203 | # Account for normalization. 204 | self.q = 4*qs 205 | # Update grid. 206 | self.grid() 207 | # Update inversion matrix. 208 | self.L = self.invmatrix(self.k, self.l) 209 | 210 | def screenlog(self): 211 | """Print model state info on screen.""" 212 | # Write time (in seconds). 213 | sys.stdout.write(' {:15.0f}'.format(self.clock)) 214 | # Write mean enstrophy for each layer. 215 | for i in range(self.nz): 216 | sys.stdout.write(' {:5e}'.format(np.mean(np.abs(self.q[:,:,i])**2) 217 | /self.n**2)) 218 | sys.stdout.write('\n') 219 | 220 | def snapshot(self, name): 221 | """Save snapshots of total q (mean added) for each level.""" 222 | # Check whether directory exists. 223 | if not os.path.isdir(name + '/snapshots'): 224 | os.makedirs(name + '/snapshots') 225 | # Transform to physical space. 226 | qp = self.irfft2(self.q, axes=(0, 1)) 227 | # Add mean gradients 228 | qp += self.qx * (self.x - self.a / 2) 229 | qp += self.qy * (self.y - self.a / 2) 230 | # Determine range of colorbars. 231 | m = np.max([np.abs(self.qx * self.a), np.abs(self.qy * self.a)], axis=0) 232 | # Save image for each layer. 233 | for i in range(self.nz): 234 | plt.imsave( 235 | name + '/snapshots/{:03d}_{:015.0f}.png'.format(i, self.clock), 236 | qp[:,:,i], origin='lower', vmin=-m[i], vmax=m[i], cmap=cm_blues) 237 | 238 | def save(self, name): 239 | """Save model state.""" 240 | # Check whether directory exists. 241 | if not os.path.isdir(name + '/data'): 242 | os.makedirs(name + '/data') 243 | # Save. 244 | with open(name + '/data/{:015.0f}'.format(self.clock), 'w') as f: 245 | pickle.dump(self, f) 246 | 247 | 248 | def load(name, time): 249 | """Load model state.""" 250 | with open(name + '/data/{:015.0f}'.format(time), 'r') as f: 251 | m = pickle.load(f) 252 | return m 253 | 254 | 255 | class TwoDim(Model): 256 | 257 | """ 258 | Two dimensional model 259 | 260 | This implements a two-dimensional model that consists of vorticity conser- 261 | vation. The inversion relation is simply a Poisson equation. 262 | """ 263 | 264 | def __init__(self): 265 | Model.__init__(self, 1) 266 | 267 | def initmean(self, qx, qy): 268 | """Set up the mean state.""" 269 | # Set up PV gradients. 270 | self.qx[0] = qx 271 | self.qy[0] = qy 272 | 273 | def invmatrix(self, k, l): 274 | """Set up the inversion matrix L.""" 275 | k2 = (k**2 + l**2)[:,:,0] 276 | k2[k2 == 0.] = 1. # preventing div. by zero for wavenumber 0 277 | L = np.empty((l.size, k.size, 1, 1)) 278 | L[:,:,0,0] = - k2 279 | return L 280 | 281 | 282 | class Surface(Model): 283 | 284 | """ 285 | Surface QG model 286 | 287 | This implements an SQG model that consists of surface buoyancy conserva- 288 | tion and implicit dynamics in an infinitely deep interior determined by 289 | zero PV there. The conserved quantity here is PV-like and implements the 290 | "oceanographic" case, where the surface is an upper surface. The dynamics 291 | can also be used in an "atmospheric" case or a case with an interface 292 | between two semi-infinite layers (see Held et al., 1995). The conserved 293 | quantity is 294 | q[0] = - f b(0) / N^2. 295 | """ 296 | 297 | def __init__(self): 298 | Model.__init__(self, 1) 299 | 300 | def initmean(self, f, N, Sx, Sy): 301 | """Set up the mean state.""" 302 | self.f = f # Coriolis parameter 303 | self.N = N # buoyancy frequency 304 | # Set up mean flow. 305 | self.u = np.array([0]) 306 | self.v = np.array([0]) 307 | # Set up mean PV gradients. 308 | self.qx = np.array([- f**2 * Sy / N**2]) 309 | self.qy = np.array([+ f**2 * Sx / N**2]) 310 | 311 | def invmatrix(self, k, l): 312 | """Set up the inversion matrix L.""" 313 | kh = np.hypot(k, l)[:,:,0] 314 | kh[kh == 0.] = 1. # preventing div. by zero for wavenumber 0 315 | L = np.empty((l.size, k.size, 1, 1)) 316 | L[:,:,0,0] = - self.f * kh / self.N 317 | return L 318 | 319 | 320 | class Layered(Model): 321 | 322 | """ 323 | Multi-layer model 324 | 325 | This implements a multi-layer model with a rigid lid. See Vallis (2006) 326 | for the formulation and notation. The layer thicknisses h, buoyancy 327 | jumps g, mean flows (u, v), and beta can be passed to initmean. 328 | """ 329 | 330 | def initmean(self, f, h, g, u, v, beta): 331 | """Set up the mean state.""" 332 | self.f = f # Coriolis parameter 333 | self.h = np.array(h) # layer thicknesses 334 | self.g = np.array(g) # buoyancy jumps at interfaces 335 | self.u = np.array(u) # mean zonal flow 336 | self.v = np.array(v) # mean meridional flow 337 | # Set up mean PV gradients. 338 | self.qx = np.zeros(self.nz) 339 | self.qx[:-1] -= f**2 * (self.v[:-1] - self.v[1:]) / (self.h[:-1] * g) 340 | self.qx[1:] += f**2 * (self.v[:-1] - self.v[1:]) / (self.h[1::] * g) 341 | self.qy = beta * np.ones(self.nz) 342 | self.qy[:-1] += f**2 * (self.u[:-1] - self.u[1:]) / (self.h[:-1] * g) 343 | self.qy[1:] -= f**2 * (self.u[:-1] - self.u[1:]) / (self.h[1::] * g) 344 | 345 | def invmatrix(self, k, l): 346 | """Set up the inversion matrix L.""" 347 | k2 = (k**2 + l**2)[:,:,0] 348 | k2[k2 == 0.] = 1. # preventing div. by zero for wavenumber 0 349 | L = np.zeros((l.size, k.size, self.nz, self.nz)) 350 | for i in range(self.nz): 351 | L[:,:,i,i] -= k2 352 | if i > 0: 353 | L[:,:,i,i-1] += self.f**2 / (self.h[i] * self.g[i-1]) 354 | L[:,:,i,i] -= self.f**2 / (self.h[i] * self.g[i-1]) 355 | if i < self.nz - 1: 356 | L[:,:,i,i] -= self.f**2 / (self.h[i] * self.g[i]) 357 | L[:,:,i,i+1] += self.f**2 / (self.h[i] * self.g[i]) 358 | return L 359 | 360 | 361 | class Eady(Model): 362 | 363 | """ 364 | Eady model 365 | 366 | This implements an Eady model that consists of surface and bottom buoyancy 367 | conservation and implicit interior dynamics determined by zero PV there. 368 | The conserved quantities here are PV-like: 369 | q[0] = - f b(0) / N^2, 370 | q[1] = + f b(-H) / N^2. 371 | """ 372 | 373 | def __init__(self): 374 | Model.__init__(self, 2) 375 | 376 | def initmean(self, f, N, H, Sx, Sy): 377 | """Set up the mean state.""" 378 | self.f = f # Coriolis parameter 379 | self.N = N # buoyancy frequency 380 | self.H = H # depth 381 | # Set up mean flow. 382 | self.u = np.array([0, - Sx * H]) 383 | self.v = np.array([0, - Sy * H]) 384 | # Set up mean PV gradients. 385 | self.qx = np.array([- f**2 * Sy / N**2, + f**2 * Sy / N**2]) 386 | self.qy = np.array([+ f**2 * Sx / N**2, - f**2 * Sx / N**2]) 387 | 388 | def invmatrix(self, k, l): 389 | """Set up the inversion matrix L.""" 390 | kh = np.hypot(k, l)[:,:,0] 391 | kh[kh == 0.] = 1. # preventing div. by zero for wavenumber 0 392 | mu = self.N * kh * self.H / self.f 393 | L = np.empty((l.size, k.size, 2, 2)) 394 | L[:,:,0,0] = - self.f * kh / (self.N * np.tanh(mu)) 395 | L[:,:,0,1] = + self.f * kh / (self.N * np.sinh(mu)) 396 | L[:,:,1,0] = + self.f * kh / (self.N * np.sinh(mu)) 397 | L[:,:,1,1] = - self.f * kh / (self.N * np.tanh(mu)) 398 | return L 399 | 400 | 401 | class FloatingEady(Model): 402 | 403 | """ 404 | Floating Eady model 405 | 406 | This implements a "floating" Eady model that consists of a layer of 407 | constant PV coupled to an infinitely deep layer below that also has 408 | constant PV (see Callies, Flierl, Ferrari, Fox-Kemper, 2015). The model 409 | consists of two conserved PV-like quantities at the surface and the inter- 410 | face between the layers: 411 | q[0] = - f b(0) / N[0]^2, 412 | q[1] = + f [b^+(-H) / N[0]^2 - b^-(-H) / N[1]^2, 413 | where N[0] and N[1] are the buoyancy frequencies of the Eady and deep 414 | layers, respectively. 415 | """ 416 | 417 | def __init__(self): 418 | Model.__init__(self, 2) 419 | 420 | def initmean(self, f, N, H, Sx, Sy): 421 | """Set up the mean state.""" 422 | self.f = f # Coriolis parameter 423 | self.N = np.array(N) # buoyancy frequencies of the two layers 424 | self.H = H # depth of upper layer 425 | # Set up mean flow. 426 | self.u = np.array([0, - Sx[0] * H]) 427 | self.v = np.array([0, - Sy[0] * H]) 428 | # Set up mean PV gradients. 429 | self.qx = np.array([ 430 | - f**2 * Sy[0] / N[0]**2, 431 | + f**2 * (Sy[0] / N[0]**2 - Sy[1] / N[1]**2)]) 432 | self.qy = np.array([ 433 | + f**2 * Sx[0] / N[0]**2, 434 | - f**2 * (Sx[0] / N[0]**2 - Sx[1] / N[1]**2)]) 435 | 436 | def invmatrix(self, k, l): 437 | """Set up the inversion matrix L.""" 438 | kh = np.hypot(k, l)[:,:,0] 439 | kh[kh == 0.] = 1. # preventing div. by zero for wavenumber 0 440 | mu = self.N[0] * kh * self.H / self.f 441 | L = np.empty((l.size, k.size, 2, 2)) 442 | L[:,:,0,0] = - self.f * kh / (self.N[0] * np.tanh(mu)) 443 | L[:,:,0,1] = + self.f * kh / (self.N[0] * np.sinh(mu)) 444 | L[:,:,1,0] = + self.f * kh / (self.N[0] * np.sinh(mu)) 445 | L[:,:,1,1] = - self.f * kh / (self.N[0] * np.tanh(mu)) \ 446 | - self.f * kh / self.N[1] 447 | return L 448 | 449 | 450 | class TwoEady(Model): 451 | 452 | """ 453 | Two-Eady model 454 | 455 | This implements a two-Eady model that consists of two layers of 456 | constant PV coupled at a deformable interface (see Callies, Flierl, 457 | Ferrari, Fox-Kemper, 2015). The model consists of three conserved PV-like 458 | quantities at the surface, the interface between the layers, and the 459 | bottom: 460 | q[0] = - f b(0) / N[0]^2, 461 | q[1] = + f [b^+(-H[0]) / N[0]^2 - b^-(-H[0]) / N[1]^2], 462 | q[0] = + f b(-H[0]-H[1]) / N[1]^2, 463 | where N[0] and N[1] are the buoyancy frequencies of the two layers and H[0] 464 | and H[1] are their depths. 465 | """ 466 | 467 | def __init__(self): 468 | Model.__init__(self, 3) 469 | 470 | def initmean(self, f, N, H, Sx, Sy): 471 | """Set up the mean state.""" 472 | self.f = f # Coriolis parameter 473 | self.N = np.array(N) # buoyancy frequencies of the two layers 474 | self.H = np.array(H) # depths of the two layers 475 | # Set up mean flow. 476 | self.u = np.array([0, - Sx[0] * H[0], - Sx[0] * H[0] - Sx[1] * H[1]]) 477 | self.v = np.array([0, - Sy[0] * H[0], - Sy[0] * H[0] - Sy[1] * H[1]]) 478 | # Set up mean PV gradients. 479 | self.qx = np.array([ 480 | - f**2 * Sy[0] / N[0]**2, 481 | + f**2 * (Sy[0] / N[0]**2 - Sy[1] / N[1]**2), 482 | + f**2 * Sy[1] / N[1]**2]) 483 | self.qy = np.array([ 484 | + f**2 * Sx[0] / N[0]**2, 485 | - f**2 * (Sx[0] / N[0]**2 - Sx[1] / N[1]**2), 486 | - f**2 * Sx[1] / N[1]**2]) 487 | 488 | def invmatrix(self, k, l): 489 | """Set up the inversion matrix L.""" 490 | kh = np.hypot(k, l) 491 | kh[kh == 0.] = 1. # preventing div. by zero for wavenumber 0 492 | mu = self.N * kh * self.H / self.f 493 | kh = kh[:,:,0] 494 | L = np.zeros((l.size, k.size, 3, 3)) 495 | L[:,:,0,0] = - self.f * kh / (self.N[0] * np.tanh(mu[:,:,0])) 496 | L[:,:,0,1] = + self.f * kh / (self.N[0] * np.sinh(mu[:,:,0])) 497 | L[:,:,1,0] = + self.f * kh / (self.N[0] * np.sinh(mu[:,:,0])) 498 | L[:,:,1,1] = - self.f * kh / (self.N[0] * np.tanh(mu[:,:,0])) \ 499 | - self.f * kh / (self.N[1] * np.tanh(mu[:,:,1])) 500 | L[:,:,1,2] = + self.f * kh / (self.N[1] * np.sinh(mu[:,:,1])) 501 | L[:,:,2,1] = + self.f * kh / (self.N[1] * np.sinh(mu[:,:,1])) 502 | L[:,:,2,2] = - self.f * kh / (self.N[1] * np.tanh(mu[:,:,1])) 503 | return L 504 | 505 | 506 | class TwoEadyJump(Model): 507 | 508 | """ 509 | Two-Eady model with jump 510 | 511 | This is a two-Eady model with an additional buoyancy and velocity jump 512 | at the interface. The model then has an additional conserved quantity, 513 | as if there was an additional layer at the interface. (The jump can be 514 | thought of as an additional, infinitesimally thin layer with infinite 515 | stratification and shear.) The conserved quantities are now 516 | q[0] = - f b(0) / N[0]^2, 517 | q[1] = + f b^+(-H[0]) / N[0]^2 + f \eta, 518 | q[1] = - f b^-(-H[0]) / N[1]^2 - f \eta, 519 | q[0] = + f b(-H[0]-H[1]) / N[1]^2, 520 | where \eta is the interface displacement. The buoyancy jump g' and the 521 | velocity jump (U, V) can be passed to initmean. 522 | """ 523 | 524 | def __init__(self): 525 | Model.__init__(self, 4) 526 | 527 | def initmean(self, f, N, H, g, Sx, Sy, U, V): 528 | """Set up the mean state.""" 529 | self.f = f # Coriolis parameter 530 | self.N = np.array(N) # buoyancy frequencies of the two layers 531 | self.H = np.array(H) # depths of the two layers 532 | self.g = g # buoyancy jump at interface 533 | # Set up mean flow. 534 | self.u = np.array([0, - Sx[0] * H[0], - Sx[0] * H[0] - U, 535 | - Sx[0] * H[0] - U - Sx[1] * H[1]]) 536 | self.v = np.array([0, - Sy[0] * H[0], - Sy[0] * H[0] - V, 537 | - Sy[0] * H[0] - V - Sy[1] * H[1]]) 538 | # Set up mean PV gradients. 539 | self.qx = np.array([ 540 | - f**2 * Sy[0] / N[0]**2, 541 | + f**2 * Sy[0] / N[0]**2 - f**2 * V / g, 542 | - f**2 * Sy[1] / N[1]**2 + f**2 * V / g, 543 | + f**2 * Sy[1] / N[1]**2]) 544 | self.qy = np.array([ 545 | + f**2 * Sx[0] / N[0]**2, 546 | - f**2 * Sx[0] / N[0]**2 + f**2 * U / g, 547 | + f**2 * Sx[1] / N[1]**2 - f**2 * U / g, 548 | - f**2 * Sx[1] / N[1]**2]) 549 | 550 | def invmatrix(self, k, l): 551 | """Set up the inversion matrix L.""" 552 | kh = np.hypot(k, l) 553 | kh[kh == 0.] = 1. # preventing div. by zero for wavenumber 0 554 | mu = self.N * kh * self.H / self.f 555 | kh = kh[:,:,0] 556 | L = np.zeros((l.size, k.size, 4, 4)) 557 | L[:,:,0,0] = - self.f * kh / (self.N[0] * np.tanh(mu[:,:,0])) 558 | L[:,:,0,1] = + self.f * kh / (self.N[0] * np.sinh(mu[:,:,0])) 559 | L[:,:,1,0] = + self.f * kh / (self.N[0] * np.sinh(mu[:,:,0])) 560 | L[:,:,1,1] = - self.f * kh / (self.N[0] * np.tanh(mu[:,:,0])) \ 561 | - self.f**2 / self.g 562 | L[:,:,1,2] = self.f**2 / self.g 563 | L[:,:,2,1] = self.f**2 / self.g 564 | L[:,:,2,2] = - self.f * kh / (self.N[1] * np.tanh(mu[:,:,1])) \ 565 | - self.f**2 / self.g 566 | L[:,:,2,3] = + self.f * kh / (self.N[1] * np.sinh(mu[:,:,1])) 567 | L[:,:,3,2] = + self.f * kh / (self.N[1] * np.sinh(mu[:,:,1])) 568 | L[:,:,3,3] = - self.f * kh / (self.N[1] * np.tanh(mu[:,:,1])) 569 | return L 570 | -------------------------------------------------------------------------------- /run.py: -------------------------------------------------------------------------------- 1 | # Copyright (C) 2013,2014,2015 Joern Callies 2 | # 3 | # This file is part of QGModel. 4 | # 5 | # QGModel is free software: you can redistribute it and/or modify 6 | # it under the terms of the GNU General Public License as published by 7 | # the Free Software Foundation, either version 3 of the License, or 8 | # (at your option) any later version. 9 | # 10 | # QGModel is distributed in the hope that it will be useful, 11 | # but WITHOUT ANY WARRANTY; without even the implied warranty of 12 | # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 13 | # GNU General Public License for more details. 14 | # 15 | # You should have received a copy of the GNU General Public License 16 | # along with QGModel. If not, see . 17 | 18 | 19 | import numpy as np 20 | import matplotlib.pyplot as plt 21 | 22 | import model 23 | 24 | 25 | # Initialization: initialize model of required type and set up mean 26 | # state. Given here are five examples, the latter three described in 27 | # Callies, Flierl, Ferrari, Fox-Kemper (2015). These are low- 28 | # resolution versions of the simulations in the paper. For the floating 29 | # Eady and two-Eady case, the resolution is insufficient to resolve the 30 | # instabilities properly. These low-resolution models can be used to 31 | # spin up higher-resolution versions (see below). The time step chosen 32 | # here may be too large for the fully turbulent regime. The model can 33 | # be restarted with a reduced time step when it blows up (see below). 34 | 35 | #folder = 'two-dim' 36 | #m = model.TwoDim() 37 | #m.initmean(0, 2e-11) 38 | 39 | #folder = 'sqg' 40 | #m = model.Surface() 41 | #m.initmean(1e-4, 8e-4, 1e-4, 0) 42 | 43 | #folder = 'two-layer' 44 | #m = model.Layered(2) 45 | #m.initmean(1e-4, 2*[250.], [1.6e-2], [2.5e-2, 0.], 2*[0.], 0.) 46 | 47 | folder = 'eady' 48 | m = model.Eady() 49 | m.initmean(1e-4, 8e-3, 500., 1e-4, 0.) 50 | 51 | #folder = 'fleady' 52 | #m = model.FloatingEady() 53 | #m.initmean(1e-4, [2e-3, 8e-3], 100., [1e-4, 1e-4], [0., 0.]) 54 | 55 | #folder = 'two-eady' 56 | #m = model.TwoEady() 57 | #m.initmean(1e-4, [2e-3, 8e-3], [100., 400.], [1e-4, 1e-4], [0., 0.]) 58 | 59 | #folder = 'two-eady-jump' 60 | #m = model.TwoEadyJump() 61 | #m.initmean(1e-4, [2e-3, 8e-3], [100., 400.], 4e-3, [1e-4, 1e-4], [0., 0.], 62 | # 2e-2, 0) 63 | 64 | # Perform linear stability analysis. This is an example of how to 65 | # calculate the linear phase speeds and growth rates for a set of 66 | # specified wavenumbers k and l for the model set up above. 67 | 68 | k = 2 * np.pi * np.logspace(-6, -3, 500) 69 | l = np.zeros(1) 70 | w = m.linstab(k, l) 71 | 72 | fig, ax = plt.subplots(2, 1, sharex=True) 73 | ax[0].semilogx(k / (2*np.pi), np.real(w[0,:,:]) / k[:,np.newaxis]) 74 | ax[1].semilogx(k / (2*np.pi), np.imag(w[0,:,:])) 75 | ax[0].set_ylabel('phase speed') 76 | ax[1].set_ylabel('growth rate') 77 | ax[1].set_xlabel('inverse wavelength') 78 | plt.show() 79 | 80 | # Set up the numerics: Here we initialize the numerics, specifying 81 | # domain size, resolution and time step. Note that initnum must always 82 | # be called after initmean has been used to set up the mean state and 83 | # model parameters, because these are used in setting up the inversion. 84 | # We also define the hyper- and hypoviscosity. In this case, high-order 85 | # hyperviscosity is used, with a coefficient that is modified from 86 | # Callies et al. (2015) to account for the the reduced resolution. 87 | 88 | m.initnum(5e5, 128, 5000.) 89 | m.initq(1e-4 * np.random.rand(128, 128, m.nz)) 90 | m.snapshot(folder) 91 | 92 | m.nu = 2.5e46 * 4.**20 93 | m.diffexp = 20 94 | m.hypodiff = 1e-16 95 | 96 | # Load model state: This simple command allows one to restart the model 97 | # from a previously saved state. All that is necessary is the folder 98 | # and the time from which the model is to be restarted. 99 | 100 | #m = model.load(folder, 50000000) 101 | 102 | # Increase resolution: This increases the model resolution and interpo- 103 | # lates the current state onto the finer grid. We also reduce the 104 | # hyperviscosity coefficient to allow more small-scale structure. 105 | 106 | #m.doubleres() 107 | #m.nu /= 2.**20 108 | 109 | # Reduce time step: This cuts the time step in half, which may be 110 | # necessary as the model becomes more energetic or the resolution is 111 | # increased. 112 | 113 | #m.dt /= 2. 114 | 115 | # Step model forward: This is the main loop of the model. It steps 116 | # forward the model, prints information on the model state, and episo- 117 | # dically saves the model state to file and a snapshot as a png-file. 118 | 119 | for i in range(20000): 120 | m.timestep() 121 | m.screenlog() 122 | if m.clock % 250000. < m.dt/10 or m.clock % 250000. - 250000. > -m.dt/10: 123 | m.save(folder) 124 | m.snapshot(folder) 125 | --------------------------------------------------------------------------------