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branches/2014/dev_r4642_WavesWG/DOC/TexFiles/Chapters/Chap_ZDF.tex
r4147 r4644 197 197 instabilities associated with too weak vertical diffusion. They must be 198 198 specified at least larger than the molecular values, and are set through 199 \np{rn\_avm0} and \np{rn\_avt0} ( namzdfnamelist, see \S\ref{ZDF_cst}).199 \np{rn\_avm0} and \np{rn\_avt0} (\ngn{namzdf} namelist, see \S\ref{ZDF_cst}). 200 200 201 201 \subsubsection{Turbulent length scale} … … 262 262 \end{equation} 263 263 264 At the ocean surface, a non zero length scale is set through the \np{rn\_lmin0} namelist 265 parameter. Usually the surface scale is given by $l_o = \kappa \,z_o$ 266 where $\kappa = 0.4$ is von Karman's constant and $z_o$ the roughness 267 parameter of the surface. Assuming $z_o=0.1$~m \citep{Craig_Banner_JPO94} 268 leads to a 0.04~m, the default value of \np{rn\_lsurf}. In the ocean interior 269 a minimum length scale is set to recover the molecular viscosity when $\bar{e}$ 270 reach its minimum value ($1.10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$ ). 264 At the ocean surface, a non zero length scale is set through the 265 \np{rn\_lmin0} namelist parameter. Usually the surface scale is given 266 by $l_o = \kappa \,z_o$ where $\kappa = 0.4$ is von Karman's constant 267 and $z_o$ the roughness parameter of the surface. Assuming $z_o=0.1$~m 268 \citep{Craig_Banner_JPO94} leads to a 0.04~m, the default value of 269 \np{rn\_lsurf}. In the ocean interior a minimum length scale is set to 270 recover the molecular viscosity when $\bar{e}$ reach its minimum value 271 ($1\times 10^{-6}= C_k\, l_{min} \,\sqrt{\bar{e}_{min}}$). 271 272 272 273 … … 283 284 \bar{e}_o = \frac{1}{2}\,\left( 15.8\,\alpha_{CB} \right)^{2/3} \,\frac{|\tau|}{\rho_o} 284 285 \end{equation} 285 where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of proportionality286 which depends on the ''wave age'', ranging from 57 for mature waves to 146 for 287 younger waves \citep{Mellor_Blumberg_JPO04}. 288 The boundary condition on the turbulent length scale follows theCharnock's relation:286 where $\alpha_{CB}$ is the \citet{Craig_Banner_JPO94} constant of 287 proportionality which depends on the ''wave age'', ranging from 57 for mature 288 waves to 146 for younger waves \citep{Mellor_Blumberg_JPO04}. The boundary 289 condition on the turbulent length scale follows Charnock's relation: 289 290 \begin{equation} \label{ZDF_Lsbc} 290 291 l_o = \kappa \beta \,\frac{|\tau|}{g\,\rho_o} 291 292 \end{equation} 292 where $\kappa=0.40$ is the von Karman constant, and $\beta$ is the Charnock's constant. 293 \citet{Mellor_Blumberg_JPO04} suggest $\beta = 2.10^{5}$ the value chosen by \citet{Stacey_JPO99} 294 citing observation evidence, and $\alpha_{CB} = 100$ the Craig and Banner's value. 295 As the surface boundary condition on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, 296 with $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}~=~67.83 corresponds 297 to $\alpha_{CB} = 100$. further setting \np{ln\_lsurf} to true applies \eqref{ZDF_Lsbc} 298 as surface boundary condition on length scale, with $\beta$ hard coded to the Stacet's value. 299 Note that a minimal threshold of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) 300 is applied on surface $\bar{e}$ value. 293 where $\kappa=0.40$ is the von Karman constant, and $\beta$ is Charnock's 294 constant. \citet{Mellor_Blumberg_JPO04} suggest $\beta = 2\times10^{5}$ the value 295 chosen by \citet{Stacey_JPO99} citing observation evidence, and $\alpha_{CB} 296 = 100$ the Craig and Banner's value. As the surface boundary condition 297 on TKE is prescribed through $\bar{e}_o = e_{bb} |\tau| / \rho_o$, with 298 $e_{bb}$ the \np{rn\_ebb} namelist parameter, setting \np{rn\_ebb}~=~67.83 299 corresponds to $\alpha_{CB} = 100$. further setting \np{ln\_lsurf} to true 300 applies \eqref{ZDF_Lsbc} as surface boundary condition on length scale, with 301 $\beta$ hard coded to Stacey's value. Note that a minimal threshold 302 of \np{rn\_emin0}$=10^{-4}~m^2.s^{-2}$ (namelist parameters) is applied on 303 surface $\bar{e}$ value. 304 305 \subsubsection{Surface wave breaking flux from a wave model \np{ln\_wavetke}} 306 %-----------------------------------------------------------------------% 307 The constant of proportionality $\alpha_{CB}$ in Eq~\eqref{ZDF_Esbc} relates 308 the water-side friction velocity $w_*$ to the turbulent energy flux as follows, 309 \begin{equation} 310 \Phi_\mathrm{oc} = \rho_\mathrm{w} \alpha_\mathrm{CB} w_*^3. 311 \label{Eq_ZDF_alpha} 312 \end{equation} 313 The default option in NEMO \eqref{ZDF_Esbc} is to assume $\alpha_{CB} 314 = 100$ as explained in the previous section. 315 However, the energy flux can be computed from the dissipation source term 316 of a wave model \citep{Janssen_AH04,Janssen_JGR12,Janssen_ECMWF13}, 317 \begin{equation} 318 \Phi_\mathrm{oc} = \Phi_\mathrm{in} - \rho_\mathrm{w}g \int_0^{2\pi} 319 \int_0^{\infty} (S_\mathrm{in} + S_\mathrm{ds})\, 320 d\omega d\theta. 321 \label{Eq_ZDF_phioc} 322 \end{equation} 323 Assuming high-frequency equilibrium and ignoring the direct turbulent energy 324 flux from the atmosphere to the ocean we get 325 \begin{equation} 326 \Phi_\mathrm{oc} = -\rho_\mathrm{w}g \int_0^{2\pi} 327 \int_0^{\omega_\mathrm{c}} S_\mathrm{ds}\, 328 d\omega d\theta = -\rho_\mathrm{a} m u_*^3. 329 \label{Eq_ZDF_m} 330 \end{equation} 331 Here, $m \approx -\sqrt{\rho_\mathrm{a}/\rho_\mathrm{w}} \alpha_\mathrm{CB}$ 332 is the energy flux \emph{from} the waves (thus always negative) normalized by 333 the air friction velocity $u_*$. It 334 is archived as PHIOC (GRIB parameter 212, table 140) in ERA-Interim and also 335 operationally by ECMWF. The namelist parameter \np{ln\_wavetke} controls 336 the wave TKE flux. We assume that the flux has been converted to physical 337 units following \eqref{Eq_ZDF_m} before ingested by NEMO. 338 339 NEMO computes the upper boundary condition following 340 \citet{Mellor_Blumberg_JPO04}, see \eqref{ZDF_Esbc}. Since $e$ varies 341 rapidly with depth, we want to weight the surface value $\overline{e}_o$ 342 by the thickness of the uppermost level to attain a value representative 343 for the turbulence level of the uppermost level, 344 \begin{equation} 345 \overline{e}_1 = \frac{\overline{e}_o}{L} \int_{-L}^{0} e(z) \,dz. 346 \label{Eq_ZDF_eavg} 347 \end{equation} 348 Here $L = \Delta z_1/2$ is the depth of the $T$-point of the uppermost level. 349 This adjustment is crucial with model configurations with a thick uppermost 350 level, e.g. ORCA1L42. If we assume, as \citet{Mellor_Blumberg_JPO04} do, that in the 351 wave-affected layer the roughness length can be set to a constant which we 352 choose to be $z_\mathrm{w} = 0.5H_\mathrm{s}$ and that in this near-surface 353 region diffusion balances dissipation, the TKE equation attains the simple 354 exponential solution \citep{Mellor_Blumberg_JPO04} 355 \begin{equation} 356 e(z) = \overline{e}_o \exp(2\lambda z/3). 357 \label{Eq_ZDF_phimb} 358 \end{equation} 359 Here, the length scale $\lambda^{-1}$ is sea-state dependent, see Eq (10) by 360 \citet{Mellor_Blumberg_JPO04}, 361 \begin{equation} 362 \lambda = [3/(S_q B_1 \kappa^2)]^{1/2}z_\mathrm{w}^{-1} \approx 363 \frac{2.38}{z_\mathrm{w}}. 364 \end{equation} 365 We have assumed $S_q=0.2$ and $B=16.6$ \citep{Mellor_Yamada_1982}, as 366 used in NEMO. For a wave height of 2.5 m, which is close to the global 367 mean, $\lambda^{-1} \approx 0.5\, \mathrm{m}$. Integrating \eqref{Eq_ZDF_phimb} 368 is straightforward, and the average TKE in \eqref{Eq_ZDF_eavg} becomes 369 \begin{equation} 370 \overline{e}_1 = \overline{e}_o \frac{3}{2\lambda L} \left[1 - 371 \exp(-2\lambda L/3)\right]. 372 \end{equation} 373 The wave model energy flux is controlled by \np{ln\_wavetke} in namelist 374 \ngn{namsbc}. 301 375 302 376 … … 318 392 319 393 By making an analogy with the characteristic convective velocity scale 320 ($e.g.$, \citet{D'Alessio_al_JPO98}), $P_{LC}$ is assumed to be 394 ($e.g.$, \citet{D'Alessio_al_JPO98}), $P_{LC}$ is assumed to be: 321 395 \begin{equation} 322 396 P_{LC}(z) = \frac{w_{LC}^3(z)}{H_{LC}}
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