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Warning, /doc/examples/deep_convection/deep_convection.rst is written in an unsupported language. File is not indexed.
view on githubraw file Latest commit 0bad585a on 2022-02-16 18:55:09 UTC1c8cebb321 Jeff*0001 .. _sec_deep_convection: d67096e55c Jeff*0002 1c8cebb321 Jeff*0003 Deep Convection 0004 =============== d67096e55c Jeff*0005 0006 (in directory: :filelink:`verification/tutorial_deep_convection/`) 0007 0008 .. figure:: figs/simulation_config.png 0009 :width: 80% 0010 :align: center 0011 :alt: deep convection setup 0012 :name: tut_deep_cvct_config 0013 0014 Schematic of simulation domain for the surface driven convection experiment. The domain is doubly periodic with an initially uniform temperature of 20 :sup:`o`\ C. 0015 0016 This experiment, :numref:`tut_deep_cvct_config`, showcasing** Warning **
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0017 MITgcm’s non-hydrostatic capability, was designed to explore the 0018 temporal and spatial characteristics of convection plumes as they might 0019 exist during a period of oceanic deep convection. It is 0020 0021 - non-hydrostatic 0022 0023 - doubly-periodic with cubic geometry 0024 0025 - discretized with 50 m resolution in :math:`x, y, z` 0026 0027 - Cartesian 0028 0029 - on an :math:`f`-plane 0030 0031 - using a linear equation of state 0032 0033 Overview 0034 -------- 0035 0036 The model domain consists of an approximately 3 km square by 1 km deep 0037 box of initially unstratified, resting fluid. The domain is doubly 0038 periodic. 0039 0040 The experiment has 20 levels in the vertical, each of equal thickness 0041 :math:`\Delta z =` 50 m (the horizontal resolution is also 50 m). The 0042 fluid is initially unstratified with a uniform reference potential 0043 temperature :math:`\theta =` 20 :sup:`o`\ C. The equation of state 0044 used in this experiment is linear 0045 0046 .. math:: 0047 \rho = \rho_{0} ( 1 - \alpha_{\theta}\theta^{'} ) 0048 :label: eg-bconv-linear1_eos 0049 0050 which is implemented in the model as a density anomaly equation 0051 0052 .. math:: 0053 \rho^{'} = -\rho_{0}\alpha_{\theta}\theta^{'} 0054 :label: eg-bconv-linear1_eos_pert 0055 0056 with :math:`\rho_{0}=1000\,{\rm kg\,m}^{-3}` and 0057 :math:`\alpha_{\theta}=2\times10^{-4}\,{\rm degrees}^{-1}`. Integrated 0058 forward in this configuration, the model state variable :varlink:`theta` is 0059 equivalent to either in-situ temperature, :math:`T`, or potential 0060 temperature, :math:`\theta`. For consistency with other examples, in 0061 which the equation of state is non-linear, we use :math:`\theta` to 0062 represent temperature here. This is the quantity that is carried in the 0063 model core equations. 0064 0065 As the fluid in the surface layer is cooled (at a mean rate of 800 0066 Wm\ :math:`^2`), it becomes convectively unstable and overturns, at 0067 first close to the grid-scale, but, as the flow matures, on larger 0068 scales (:numref:`tut_deep_cvct_vert_section` and 0069 :numref:`tut_deep_cvct_surf_section`), under the influence of rotation 0070 (:math:`f_o = 10^{-4}` s\ :math:`^{-1}`). 0071 0072 .. figure:: figs/verticalsection.png 0073 :width: 80% 0074 :align: center 0075 :alt: vertical section deep cvct exp 0076 :name: tut_deep_cvct_vert_section 0077 0078 Vertical section 0079 0080 .. figure:: figs/surfacesection.png 0081 :width: 80% 0082 :align: center 0083 :alt: surface section deep cvct exp 0084 :name: tut_deep_cvct_surf_section 0085 0086 Surface section 0087 0088 Model parameters are specified in file :filelink:`input/data <verification/tutorial_deep_convection/input/data>`. The grid dimensions 0089 are prescribed in :filelink:`code/SIZE.h <verification/tutorial_deep_convection/code/SIZE.h>`. The forcing (file ``input/Qsurf.bin``) is 0090 specified in a binary data file generated using the Matlab script 0091 :filelink:`input/gendata.m <verification/tutorial_deep_convection/input/gendata.m>`. 0092 0093 Equations solved 0094 ---------------- 0095 0096 The model is configured in non-hydrostatic form, that is, all terms in 0097 the Navier Stokes equations are retained and the pressure field is 0098 found, subject to appropriate boundary conditions, through inversion of 0099 a 3-D elliptic equation. 0100 0101 The implicit free surface form of the pressure equation described in 0102 Marshall et. al (1997) :cite:`marshall:97a` is employed. A 0103 horizontal Laplacian operator :math:`\nabla_{h}^2` provides viscous 0104 dissipation. The thermodynamic forcing appears as a sink in the 0105 equation for potential temperature :math:`\theta`. This produces a set of equations 0106 solved in this configuration as follows: 0107 0108 .. math:: 0109 0110 \begin{aligned} 0111 \frac{Du}{Dt} - fv + 0112 \frac{1}{\rho}\frac{\partial p^{'}}{\partial x} - 0113 \nabla_{h}\cdot A_{h}\nabla_{h}u - 0114 \frac{\partial}{\partial z}A_{z}\frac{\partial u}{\partial z} 0bad585a21 Navi*0115 & = d67096e55c Jeff*0116 \begin{cases} 0117 0 & \text{(surface)} \\ 0118 0 & \text{(interior)} 0119 \end{cases} 0120 \\ 0121 \frac{Dv}{Dt} + fu + 0122 \frac{1}{\rho}\frac{\partial p^{'}}{\partial y} - 0123 \nabla_{h}\cdot A_{h}\nabla_{h}v - 0124 \frac{\partial}{\partial z}A_{z}\frac{\partial v}{\partial z} 0bad585a21 Navi*0125 & = d67096e55c Jeff*0126 \begin{cases} 0127 0 & \text{(surface)} \\ 0128 0 & \text{(interior)} 0129 \end{cases} 0130 \\ 0131 \frac{Dw}{Dt} + g \frac{\rho^{'}}{\rho} + 0132 \frac{1}{\rho}\frac{\partial p^{'}}{\partial z} - 0133 \nabla_{h}\cdot A_{h}\nabla_{h}w - 0134 \frac{\partial}{\partial z}A_{z}\frac{\partial w}{\partial z} 0bad585a21 Navi*0135 & = d67096e55c Jeff*0136 \begin{cases} 0137 0 & \text{(surface)} \\ 0138 0 & \text{(interior)} 0139 \end{cases} 0140 \\ 0141 \frac{\partial u}{\partial x} + 0142 \frac{\partial v}{\partial y} + 0143 \frac{\partial w}{\partial z} + 0bad585a21 Navi*0144 &= d67096e55c Jeff*0145 0 0146 \\ 0147 \frac{D\theta}{Dt} - 0148 \nabla_{h}\cdot K_{h}\nabla_{h}\theta 0149 - \frac{\partial}{\partial z}K_{z}\frac{\partial\theta}{\partial z} 0bad585a21 Navi*0150 & = d67096e55c Jeff*0151 \begin{cases} 0152 {\cal F}_\theta & \text{(surface)} \\ 0153 0 & \text{(interior)} 0154 \end{cases} 0155 \end{aligned} 0156 0157 where :math:`u=\frac{Dx}{Dt}`, :math:`v=\frac{Dy}{Dt}` and 0158 :math:`w=\frac{Dz}{Dt}` are the components of the flow vector in 0159 directions :math:`x`, :math:`y` and :math:`z`. The pressure is 0160 diagnosed through inversion (subject to appropriate boundary 0161 conditions) of a 3-D elliptic equation derived from the divergence of 0162 the momentum equations and continuity (see :numref:`finding_the_pressure_field`). 0163 0164 Discrete numerical configuration 0165 -------------------------------- 0166 0167 The domain is discretized with a uniform grid spacing in each direction. 0168 There are 64 grid cells in directions :math:`x` and :math:`y` and 20 0169 vertical levels thus the domain comprises a total of just over 80,000 0170 gridpoints. 0171 0172 Numerical stability criteria and other considerations 0173 ----------------------------------------------------- 0174 0175 For a heat flux of 800 Wm\ :math:`^2` and a rotation rate of 0176 :math:`10^{-4}` s\ :math:`^{-1}` the plume-scale can be expected to be a 0177 few hundred meters guiding our choice of grid resolution. This in turn 0178 restricts the timestep we can take. It is also desirable to minimize the 0179 level of diffusion and viscosity we apply. 0180 0181 For this class of problem it is generally the advective time-scale which 0182 restricts the timestep. 0183 0184 For an extreme maximum flow speed of :math:`| \vec{u} | = 1 ms^{-1}`, 0185 at a resolution of 50 m, the implied maximum timestep for stability, 0186 :math:`\delta t_u` is 0187 0188 .. math:: 0189 \delta t_u = \frac{\Delta x}{| \vec{u} |} = 50 s 0190 0191 The choice of :math:`\delta t = 10` s is a safe 20 percent of this 0192 maximum. 0193 0194 Interpreted in terms of a mixing-length hypothesis, a magnitude of 0195 Laplacian diffusion coefficient :math:`\kappa_h (= 0196 \kappa_v) = 0.1` m\ :math:`^2`\ s\ :math:`^{-1}` is consistent with an 0197 eddy velocity of 2 mm s\ :math:`^{-1}` correlated over 50 m. 0198 0199 Experiment configuration 0200 ------------------------ 0201 0202 The model configuration for this experiment resides under the directory 0203 *verification/convection/*. The experiment files 0204 0205 - :filelink:`code/CPP_OPTIONS.h <verification/tutorial_deep_convection/code/CPP_OPTIONS.h>` 0206 - :filelink:`code/SIZE.h <verification/tutorial_deep_convection/code/SIZE.h>` 0207 - :filelink:`input/data <verification/tutorial_deep_convection/input/data>` 0208 - :filelink:`input/data.pkg <verification/tutorial_deep_convection/input/data.pkg>` 0209 - :filelink:`input/eedata <verification/tutorial_deep_convection/input/eedata>` 0210 - ``input/Qsurf.bin``, 0211 0212 contain the code customizations and parameter settings for this 0213 experiment. Below we describe these experiment-specific customizations. 0214 0215 File :filelink:`code/CPP_OPTIONS.h <verification/tutorial_deep_convection/code/CPP_OPTIONS.h>` 0216 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0217 0218 This file uses standard default values and does not contain 0219 customizations for this experiment. 0220 0221 File :filelink:`code/SIZE.h <verification/tutorial_deep_convection/code/SIZE.h>` 0222 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0223 0224 .. literalinclude:: ../../../verification/tutorial_deep_convection/code/SIZE.h 0225 :linenos: 0226 :caption: verification/tutorial_deep_convection/code/SIZE.h 0227 0228 Three lines are customized in this file. These prescribe the domain grid 0229 dimensions. 0230 0231 - Line 45, 0232 0233 :: 0234 0235 sNx=50, 0236 0237 this line sets the lateral domain extent in grid points for the axis 0238 aligned with the :math:`x`-coordinate. 0239 0240 - Line 46, 0241 0242 :: 0243 0244 sNy=50, 0245 0246 this line sets the lateral domain extent in grid points for the axis 0247 aligned with the :math:`y`-coordinate. 0248 0249 - Line 55, 0250 0251 :: 0252 0253 Nr=50, 0254 0255 this line sets the vertical domain extent in grid points. 0256 0257 File :filelink:`input/data <verification/tutorial_deep_convection/input/data>` 0258 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0259 0260 .. literalinclude:: ../../../verification/tutorial_deep_convection/input/data 0261 :linenos: 0262 :caption: verification/tutorial_deep_convection/input/data 0263 0264 This file specifies the main parameters 0265 for the experiment. The parameters that are significant for this 0266 configuration are 0267 0268 - Line 7, 0269 0270 :: 0271 0272 tRef=20*20.0, 0273 0274 this line sets the initial and reference values of potential 0275 temperature at each model level in units of 0276 :math:`^{\circ}\mathrm{C}`. Here the value is arbitrary since, in 0277 this case, the flow evolves independently of the absolute magnitude 0278 of the reference temperature. For each depth level the initial and 0279 reference profiles will be uniform in :math:`x` and :math:`y`. 0280 0281 - Line 8, 0282 0283 :: 0284 0285 sRef=20*35.0, 0286 0287 this line sets the initial and reference values of salinity at each 0288 model level in units of ppt. In this case salinity is set to an 0289 (arbitrary) uniform value of 35.0 ppt. However since, in this 0290 example, density is independent of salinity, an appropriately defined 0291 initial salinity could provide a useful passive tracer. For each 0292 depth level the initial and reference profiles will be uniform in 0293 :math:`x` and :math:`y`. 0294 0295 - Line 9, 0296 0297 :: 0298 0299 viscAh=0.1, 0300 0301 this line sets the horizontal Laplacian dissipation coefficient to 0302 0.1 :math:`{\rm m^{2}s^{-1}}`. Boundary conditions for this operator 0303 are specified later. 0304 0305 - Line 10, 0306 0307 :: 0308 0309 viscAz=0.1, 0310 0311 this line sets the vertical Laplacian frictional dissipation 0312 coefficient to 0.1 :math:`{\rm m^{2}s^{-1}}`. Boundary conditions for 0313 this operator are specified later. 0314 0315 - Line 11, 0316 0317 :: 0318 0319 no_slip_sides=.FALSE. 0320 0321 this line selects a free-slip lateral boundary condition for the 0322 horizontal Laplacian friction operator e.g. 0323 :math:`\frac{\partial u}{\partial y}`\ =0 along boundaries in 0324 :math:`y` and :math:`\frac{\partial v}{\partial x}`\ =0 along 0325 boundaries in :math:`x`. 0326 0327 - Lines 12, 0328 0329 :: 0330 0331 no_slip_bottom=.TRUE. 0332 0333 this line selects a no-slip boundary condition for the bottom 0334 boundary condition in the vertical Laplacian friction operator e.g., 0335 :math:`u=v=0` at :math:`z=-H`, where :math:`H` is the local depth of 0336 the domain. 0337 0338 - Line 13, 0339 0340 :: 0341 0342 diffKhT=0.1, 0343 0344 this line sets the horizontal diffusion coefficient for temperature 0345 to 0.1 :math:`\rm m^{2}s^{-1}`. The boundary condition on this 0346 operator is 0347 :math:`\frac{\partial}{\partial x}=\frac{\partial}{\partial y}=0` at 0348 all boundaries. 0349 0350 - Line 14, 0351 0352 :: 0353 0354 diffKzT=0.1, 0355 0356 this line sets the vertical diffusion coefficient for temperature to 0357 0.1 :math:`{\rm m^{2}s^{-1}}`. The boundary condition on this 0358 operator is :math:`\frac{\partial}{\partial z}` = 0 on all 0359 boundaries. 0360 0361 - Line 15, 0362 0363 :: 0364 0365 f0=1E-4, 0366 0367 this line sets the Coriolis parameter to :math:`1 \times 10^{-4}` 0368 s\ :math:`^{-1}`. Since :math:`\beta = 0.0` this value is then 0369 adopted throughout the domain. 0370 0371 - Line 16, 0372 0373 :: 0374 0375 beta=0.E-11, 0376 0377 this line sets the the variation of Coriolis parameter with latitude 0378 to :math:`0`. 0379 0380 - Line 17, 0381 0382 :: 0383 0384 tAlpha=2.E-4, 0385 0386 This line sets the thermal expansion coefficient for the fluid to 0387 :math:`2 \times 10^{-4}` :sup:`o`\ C\ :math:`^{-1}`. 0388 0389 - Line 18, 0390 0391 :: 0392 0393 sBeta=0, 0394 0395 This line sets the saline expansion coefficient for the fluid to** Warning **
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0396 :math:`0`, consistent with salt’s passive role in this example. 0397 0398 - Line 23-24, 0399 0400 :: 0401 0402 rigidLid=.FALSE., 0403 implicitFreeSurface=.TRUE., 0404 0405 Selects the barotropic pressure equation to be the implicit free 0406 surface formulation. 0407 0408 - Line 26, 0409 0410 :: 0411 0412 eosType='LINEAR', 0413 0414 Selects the linear form of the equation of state. 0415 0416 - Line 27, 0417 0418 :: 0419 0420 nonHydrostatic=.TRUE., 0421 0422 Selects for non-hydrostatic code. 0423 0424 - Line 33, 0425 0426 :: 0427 0428 cg2dMaxIters=1000, 0429 0430 Inactive - the pressure field in a non-hydrostatic simulation is 0431 inverted through a 3-D elliptic equation. 0432 0433 - Line 34, 0434 0435 :: 0436 0437 cg2dTargetResidual=1.E-9, 0438 0439 Inactive - the pressure field in a non-hydrostatic simulation is 0440 inverted through a 3-D elliptic equation. 0441 0442 - Line 35, 0443 0444 :: 0445 0446 cg3dMaxIters=40, 0447 0448 This line sets the maximum number of iterations the 0449 3-D conjugate gradient solver will use to 40, 0450 **irrespective of the convergence criteria being met**. 0451 0452 - Line 36, 0453 0454 :: 0455 0456 cg3dTargetResidual=1.E-9, 0457 0458 Sets the tolerance which the 3-D conjugate gradient 0459 solver will use to test for convergence in equation :eq:`phi-nh` to 0460 :math:`1 \times 10^{-9}`. The solver will iterate until the tolerance 0461 falls below this value or until the maximum number of solver 0462 iterations is reached. 0463 0464 - Line43, 0465 0466 :: 0467 0468 nTimeSteps=8640., 0469 0470 Sets the number of timesteps at which this simulation will terminate 0471 (in this case 8640 timesteps or 1 day or :math:`\delta t = 10` s). At 0472 the end of a simulation a checkpoint file is automatically written so 0473 that a numerical experiment can consist of multiple stages. 0474 0475 - Line 44, 0476 0477 :: 0478 0479 deltaT=10, 0480 0481 Sets the timestep :math:`\delta t` to 10 s. 0482 0483 - Line 57, 0484 0485 :: 0486 0487 dXspacing=50.0, 0488 0489 Sets horizontal (:math:`x`-direction) grid interval to 50 m. 0490 0491 - Line 58, 0492 0493 :: 0494 0495 dYspacing=50.0, 0496 0497 Sets horizontal (:math:`y`-direction) grid interval to 50 m. 0498 0499 - Line 59, 0500 0501 :: 0502 0503 delZ=20*50.0, 0504 0505 Sets vertical grid spacing to 50 m. Must be consistent with 0506 :filelink:`code/SIZE.h <verification/tutorial_deep_convection/code/SIZE.h>`. 0507 Here, 20 corresponds to the number of vertical levels. 0508 0509 - Line64, 0510 0511 :: 0512 0513 surfQfile='Qsurf.bin' 0514 0515 This line specifies the name of the file from which the surface heat 0516 flux is read. This file is a 2-D (:math:`x,y`) map. It is 0517 assumed to contain 64-bit binary numbers giving the value of 0518 :math:`Q` (W m\ :math:`^2`) to be applied in each surface grid cell, 0519 ordered with the :math:`x` coordinate varying fastest. The points are 0520 ordered from low coordinate to high coordinate for both axes. The 0521 matlab program :filelink:`input/gendata.m <verification/tutorial_deep_convection/input/gendata.m>` 0522 shows how to generate the surface 0523 heat flux file used in the example. 0524 0525 File :filelink:`input/data.pkg <verification/tutorial_deep_convection/input/data.pkg>` 0526 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0527 0528 This file uses standard default values and does not contain 0529 customizations for this experiment. 0530 0531 File :filelink:`input/eedata <verification/tutorial_deep_convection/input/eedata>` 0532 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 0533 0534 This file uses standard default values and does not contain 0535 customizations for this experiment. 0536 0537 File ``input/Qsurf.bin`` 0538 ~~~~~~~~~~~~~~~~~~~~~~~~ 0539 0540 The file ``input/Qsurf.bin`` specifies a 2-D (:math:`x,y`) map 0541 of heat flux values where 0542 :math:`Q = Q_o \times ( 0.5 + \mbox{random number between 0 and 1})`. 0543 0544 In the example :math:`Q_o = 800` W m\ :math:`^{-2}` so that values of 0545 :math:`Q` lie in the range 400 to 1200 W m\ :math:`^{-2}` with a mean of 0546 :math:`Q_o`. Although the flux models a loss, because it is directed** Warning **
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0547 upwards, according to the model’s sign convention, :math:`Q` is 0548 positive. 0549
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