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r9414 r10354 22 22 \label{sec:C.0} 23 23 24 Notation used in this appendix in the demonstations 24 Notation used in this appendix in the demonstations: 25 25 26 26 fluxes at the faces of a $T$-box: … … 37 37 38 38 $dv=e_1\,e_2\,e_3 \,di\,dj\,dk$ is the volume element, with only $e_3$ that depends on time. 39 $D$ and $S$ are the ocean domain volume and surface, respectively. 40 No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$) 41 Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. 39 $D$ and $S$ are the ocean domain volume and surface, respectively. 40 No wetting/drying is allow ($i.e.$ $\frac{\partial S}{\partial t} = 0$). 41 Let $k_s$ and $k_b$ be the ocean surface and bottom, resp. 42 42 ($i.e.$ $s(k_s) = \eta$ and $s(k_b)=-H$, where $H$ is the bottom depth). 43 43 \begin{flalign*} … … 60 60 = \int_D { \frac{1}{e_3} \partial_t \left( e_3 \, Q \right) dv } =0 61 61 \end{equation*} 62 equation of evolution of $Q$ written as the time evolution of the vertical content of $Q$ 63 like for tracers, or momentum in flux form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when : 62 equation of evolution of $Q$ written as 63 the time evolution of the vertical content of $Q$ like for tracers, or momentum in flux form, 64 the quadratic quantity $\frac{1}{2}Q^2$ is conserved when: 64 65 \begin{flalign*} 65 66 \partial_t \left( \int_D{ \frac{1}{2} \,Q^2\;dv } \right) … … 74 75 - \frac{1}{2} \int_D { \frac{Q^2}{e_3} \partial_t (e_3) \;dv } 75 76 \end{flalign} 76 equation of evolution of $Q$ written as the time evolution of $Q$ 77 like for momentum in vector invariant form, the quadratic quantity $\frac{1}{2}Q^2$ is conserved when:77 equation of evolution of $Q$ written as the time evolution of $Q$ like for momentum in vector invariant form, 78 the quadratic quantity $\frac{1}{2}Q^2$ is conserved when: 78 79 \begin{flalign*} 79 80 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) … … 82 83 + \int_D { \frac{1}{2} Q^2 \, \partial_t e_3 \;e_1e_2\;di\,dj\,dk } \\ 83 84 \end{flalign*} 84 that is in a more compact form 85 that is in a more compact form: 85 86 \begin{flalign} \label{eq:Q2_vect} 86 87 \partial_t \left( \int_D {\frac{1}{2} Q^2\;dv} \right) … … 97 98 98 99 99 The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying 100 vertical coordinate) must be chosen so that the discrete equation of the model satisfy 101 integral constrains on energy and enstrophy. 100 The discretization of pimitive equation in $s$-coordinate ($i.e.$ time and space varying vertical coordinate) 101 must be chosen so that the discrete equation of the model satisfy integral constrains on energy and enstrophy. 102 102 103 103 104 104 Let us first establish those constraint in the continuous world. 105 The total energy ($i.e.$ kinetic plus potential energies) is conserved 105 The total energy ($i.e.$ kinetic plus potential energies) is conserved: 106 106 \begin{flalign} \label{eq:Tot_Energy} 107 107 \partial_t \left( \int_D \left( \frac{1}{2} {\textbf{U}_h}^2 + \rho \, g \, z \right) \;dv \right) = & 0 108 108 \end{flalign} 109 under the following assumptions: no dissipation, no forcing 110 (wind, buoyancy flux, atmospheric pressure variations), mass 111 conservation, and closed domain. 112 113 This equation can be transformed to obtain several sub-equalities. 114 The transformation for the advection term depends on whether 115 the vector invariant form or the flux form is used for the momentum equation. 116 Using \autoref{eq:Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in \autoref{eq:Tot_Energy} 117 for the former form and 118 Using \autoref{eq:Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in \autoref{eq:Tot_Energy} 119 for the latter form leads to: 109 under the following assumptions: no dissipation, no forcing (wind, buoyancy flux, atmospheric pressure variations), 110 mass conservation, and closed domain. 111 112 This equation can be transformed to obtain several sub-equalities. 113 The transformation for the advection term depends on whether the vector invariant form or 114 the flux form is used for the momentum equation. 115 Using \autoref{eq:Q2_vect} and introducing \autoref{apdx:A_dyn_vect} in 116 \autoref{eq:Tot_Energy} for the former form and 117 using \autoref{eq:Q2_flux} and introducing \autoref{apdx:A_dyn_flux} in 118 \autoref{eq:Tot_Energy} for the latter form leads to: 120 119 121 120 \begin{subequations} \label{eq:E_tot} … … 348 347 349 348 Substituting the discrete expression of the time derivative of the velocity either in vector invariant, 350 leads to the discrete equivalent of the four equations \autoref{eq:E_tot_flux}. 349 leads to the discrete equivalent of the four equations \autoref{eq:E_tot_flux}. 351 350 352 351 % ------------------------------------------------------------------------------------------------------------- … … 356 355 \label{subsec:C_vor} 357 356 358 Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), or 359 the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). 360 Two discretisation of the vorticity term (ENE and EEN) allows the conservation of 361 the kinetic energy. 357 Let $q$, located at $f$-points, be either the relative ($q=\zeta / e_{3f}$), 358 or the planetary ($q=f/e_{3f}$), or the total potential vorticity ($q=(\zeta +f) /e_{3f}$). 359 Two discretisation of the vorticity term (ENE and EEN) allows the conservation of the kinetic energy. 362 360 % ------------------------------------------------------------------------------------------------------------- 363 361 % Vorticity Term with ENE scheme … … 366 364 \label{subsec:C_vorENE} 367 365 368 For the ENE scheme, the two components of the vorticity term are given by 366 For the ENE scheme, the two components of the vorticity term are given by: 369 367 \begin{equation*} 370 368 - e_3 \, q \;{\textbf{k}}\times {\textbf {U}}_h \equiv … … 377 375 \end{equation*} 378 376 379 This formulation does not conserve the enstrophy but it does conserve the 380 total kinetic energy. Indeed, the kinetic energy tendency associated to the 381 vorticity term and averaged over the ocean domain can be transformed as 382 follows: 377 This formulation does not conserve the enstrophy but it does conserve the total kinetic energy. 378 Indeed, the kinetic energy tendency associated to the vorticity term and 379 averaged over the ocean domain can be transformed as follows: 383 380 \begin{flalign*} 384 381 &\int\limits_D - \left( e_3 \, q \;\textbf{k} \times \textbf{U}_h \right) \cdot \textbf{U}_h \; dv && \\ … … 412 409 \end{aligned} } \right. 413 410 \end{equation} 414 where the indices $i_p$ and $j_p$ take the following value: 415 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 411 where the indices $i_p$ and $j_p$ take the following value: $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 416 412 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: 417 413 \begin{equation} \tag{\ref{eq:Q_triads}} … … 420 416 \end{equation} 421 417 422 This formulation does conserve the total kinetic energy. Indeed, 418 This formulation does conserve the total kinetic energy. 419 Indeed, 423 420 \begin{flalign*} 424 421 &\int\limits_D - \textbf{U}_h \cdot \left( \zeta \;\textbf{k} \times \textbf{U}_h \right) \; dv && \\ … … 473 470 \label{subsec:C_zad} 474 471 475 The change of Kinetic Energy (KE) due to the vertical advection is exactly 476 balanced by the change of KE due to the horizontal gradient of KE~: 472 The change of Kinetic Energy (KE) due to the vertical advection is exactly balanced by the change of KE due to the horizontal gradient of KE~: 477 473 \begin{equation*} 478 474 \int_D \textbf{U}_h \cdot \frac{1}{e_3 } \omega \partial_k \textbf{U}_h \;dv … … 480 476 + \frac{1}{2} \int_D { \frac{{\textbf{U}_h}^2}{e_3} \partial_t ( e_3) \;dv } \\ 481 477 \end{equation*} 482 Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry 483 property of the $\delta$ operator) and the continuity equation, then 484 \autoref{eq:DOM_di_adj} again, then the commutativity of operators 485 $\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj} 486 ($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator) 478 Indeed, using successively \autoref{eq:DOM_di_adj} ($i.e.$ the skew symmetry property of the $\delta$ operator) 479 and the continuity equation, then \autoref{eq:DOM_di_adj} again, 480 then the commutativity of operators $\overline {\,\cdot \,}$ and $\delta$, and finally \autoref{eq:DOM_mi_adj} 481 ($i.e.$ the symmetry property of the $\overline {\,\cdot \,}$ operator) 487 482 applied in the horizontal and vertical directions, it becomes: 488 483 \begin{flalign*} … … 543 538 \end{flalign*} 544 539 545 There is two main points here. First, the satisfaction of this property links the choice of 546 the discrete formulation of the vertical advection and of the horizontal gradient 547 of KE. Choosing one imposes the other. For example KE can also be discretized 548 as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. This leads to the following 549 expression for the vertical advection: 540 There is two main points here. 541 First, the satisfaction of this property links the choice of the discrete formulation of the vertical advection and 542 of the horizontal gradient of KE. 543 Choosing one imposes the other. 544 For example KE can also be discretized as $1/2\,({\overline u^{\,i}}^2 + {\overline v^{\,j}}^2)$. 545 This leads to the following expression for the vertical advection: 550 546 \begin{equation*} 551 547 \frac{1} {e_3 }\; \omega\; \partial_k \textbf{U}_h … … 557 553 \end{array}} } \right) 558 554 \end{equation*} 559 a formulation that requires an additional horizontal mean in contrast with 560 the one used in NEMO. Nine velocity points have to be used instead of 3. 555 a formulation that requires an additional horizontal mean in contrast with the one used in NEMO. 556 Nine velocity points have to be used instead of 3. 561 557 This is the reason why it has not been chosen. 562 558 563 Second, as soon as the chosen $s$-coordinate depends on time, an extra constraint564 a rises on the time derivative of the volume at $u$- and $v$-points:559 Second, as soon as the chosen $s$-coordinate depends on time, 560 an extra constraint arises on the time derivative of the volume at $u$- and $v$-points: 565 561 \begin{flalign*} 566 562 e_{1u}\,e_{2u}\,\partial_t (e_{3u}) =\overline{ e_{1t}\,e_{2t}\;\partial_t (e_{3t}) }^{\,i+1/2} \\ … … 583 579 584 580 \gmcomment{ 585 A pressure gradient has no contribution to the evolution of the vorticity as the 586 curl of a gradient is zero. In the $z$-coordinate, this property is satisfied locally 587 on a C-grid with 2nd order finite differences(property \autoref{eq:DOM_curl_grad}).581 A pressure gradient has no contribution to the evolution of the vorticity as the curl of a gradient is zero. 582 In the $z$-coordinate, this property is satisfied locally on a C-grid with 2nd order finite differences 583 (property \autoref{eq:DOM_curl_grad}). 588 584 } 589 585 590 When the equation of state is linear ($i.e.$ when an advection-diffusion equation591 for density can be derived from those of temperature and salinity) the change of 592 KE due to the work of pressure forces is balanced by the change of potential 593 energy due to buoyancy forces:586 When the equation of state is linear 587 ($i.e.$ when an advection-diffusion equation for density can be derived from those of temperature and salinity) 588 the change of KE due to the work of pressure forces is balanced by 589 the change of potential energy due to buoyancy forces: 594 590 \begin{equation*} 595 591 - \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv … … 598 594 \end{equation*} 599 595 600 This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. 601 Indeed, defining the depth of a $T$-point, $z_t$, as the sum of the vertical scale602 factors at $w$-points starting from the surface, the work of pressure forces can be 603 written as:596 This property can be satisfied in a discrete sense for both $z$- and $s$-coordinates. 597 Indeed, defining the depth of a $T$-point, $z_t$, 598 as the sum of the vertical scale factors at $w$-points starting from the surface, 599 the work of pressure forces can be written as: 604 600 \begin{flalign*} 605 601 &- \int_D \left. \nabla p \right|_z \cdot \textbf{U}_h \;dv … … 658 654 \end{flalign*} 659 655 The first term is exactly the first term of the right-hand-side of \autoref{eq:KE+PE_vect_discrete}. 660 It remains to demonstrate that the last term, which is obviously a discrete analogue of 661 $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to the last term of \autoref{eq:KE+PE_vect_discrete}. 656 It remains to demonstrate that the last term, 657 which is obviously a discrete analogue of $\int_D \frac{p}{e_3} \partial_t (e_3)\;dv$ is equal to 658 the last term of \autoref{eq:KE+PE_vect_discrete}. 662 659 In other words, the following property must be satisfied: 663 660 \begin{flalign*} … … 666 663 \end{flalign*} 667 664 668 Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$. 665 Let introduce $p_w$ the pressure at $w$-point such that $\delta_k [p_w] = - \rho \,g\,e_{3t}$. 669 666 The right-hand-side of the above equation can be transformed as follows: 670 667 … … 718 715 719 716 720 Note that this property strongly constrains the discrete expression of both 721 the depth of $T-$points and of the term added to the pressure gradient in the 722 $s$-coordinate. Nevertheless, it is almost never satisfied since a linear equation 723 of state is rarely used. 717 Note that this property strongly constrains the discrete expression of both the depth of $T-$points and 718 of the term added to the pressure gradient in the $s$-coordinate. 719 Nevertheless, it is almost never satisfied since a linear equation of state is rarely used. 724 720 725 721 … … 755 751 \end{flalign*} 756 752 757 Substituting the discrete expression of the time derivative of the velocity either in vector invariant or in flux form,758 leads to the discrete equivalent of the 753 Substituting the discrete expression of the time derivative of the velocity either in 754 vector invariant or in flux form, leads to the discrete equivalent of the ???? 759 755 760 756 … … 771 767 \label{subsec:C.3.3} 772 768 773 In flux from the vorticity term reduces to a Coriolis term in which the Coriolis 774 parameter has been modified to account for the ``metric'' term. This altered 775 Coriolis parameter is discretised at an f-point. It is given by: 769 In flux from the vorticity term reduces to a Coriolis term in which 770 the Coriolis parameter has been modified to account for the ``metric'' term. 771 This altered Coriolis parameter is discretised at an f-point. 772 It is given by: 776 773 \begin{equation*} 777 774 f+\frac{1} {e_1 e_2 } \left( v \frac{\partial e_2 } {\partial i} - u \frac{\partial e_1 } {\partial j}\right)\; … … 781 778 \end{equation*} 782 779 783 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 784 It therefore conserves the total KE. The derivation is the same as for the785 vorticity term in the vector invariant form (\autoref{subsec:C_vor}).780 Either the ENE or EEN scheme is then applied to obtain the vorticity term in flux form. 781 It therefore conserves the total KE. 782 The derivation is the same as for the vorticity term in the vector invariant form (\autoref{subsec:C_vor}). 786 783 787 784 % ------------------------------------------------------------------------------------------------------------- … … 791 788 \label{subsec:C.3.4} 792 789 793 The flux form operator of the momentum advection is evaluated using a 794 centered second order finite difference scheme. Because of the flux form, 795 the discrete operator does not contribute to the global budget of linear 796 momentum. Because of the centered second order scheme, it conserves 797 the horizontal kinetic energy, that is : 790 The flux form operator of the momentum advection is evaluated using 791 a centered second order finite difference scheme. 792 Because of the flux form, the discrete operator does not contribute to the global budget of linear momentum. 793 Because of the centered second order scheme, it conserves the horizontal kinetic energy, that is: 798 794 799 795 \begin{equation} \label{eq:C_ADV_KE_flux} … … 804 800 \end{equation} 805 801 806 Let us first consider the first term of the scalar product ($i.e.$ just the the terms807 associated with the i-component of the advection):802 Let us first consider the first term of the scalar product 803 ($i.e.$ just the the terms associated with the i-component of the advection): 808 804 \begin{flalign*} 809 805 & - \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv \\ … … 845 841 \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} &&& \\ 846 842 \end{flalign*} 847 Applying similar manipulation applied to the second term of the scalar product 848 leads to : 843 Applying similar manipulation applied to the second term of the scalar product leads to: 849 844 \begin{equation*} 850 845 - \int_D \textbf{U}_h \cdot \left( {{\begin{array} {*{20}c} … … 854 849 \biggl\{ \left( \frac{1}{e_{3t}} \frac{\partial e_{3t}}{\partial t} \right) \; b_t \biggr\} 855 850 \end{equation*} 856 which is the discrete form of 857 $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. 851 which is the discrete form of $ \frac{1}{2} \int_D u \cdot \nabla \cdot \left( \textbf{U}\,u \right) \; dv $. 858 852 \autoref{eq:C_ADV_KE_flux} is thus satisfied. 859 853 860 854 861 When the UBS scheme is used to evaluate the flux form momentum advection, 862 the discrete operator does not contribute to the global budget of linear momentum 863 (flux form). The horizontal kinetic energy is not conserved, but forced to decay 864 ($i.e.$ the scheme is diffusive). 855 When the UBS scheme is used to evaluate the flux form momentum advection, 856 the discrete operator does not contribute to the global budget of linear momentum (flux form). 857 The horizontal kinetic energy is not conserved, but forced to decay ($i.e.$ the scheme is diffusive). 865 858 866 859 … … 894 887 \end{equation} 895 888 896 The scheme does not allow but the conservation of the total kinetic energy but the conservation 897 of $q^2$, the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$). 898 Indeed, using the symmetry or skew symmetry properties of the operators ( \autoref{eq:DOM_mi_adj} 899 and \autoref{eq:DOM_di_adj}), it can be shown that: 889 The scheme does not allow but the conservation of the total kinetic energy but the conservation of $q^2$, 890 the potential enstrophy for a horizontally non-divergent flow ($i.e.$ when $\chi$=$0$). 891 Indeed, using the symmetry or skew symmetry properties of the operators 892 ( \autoref{eq:DOM_mi_adj} and \autoref{eq:DOM_di_adj}), 893 it can be shown that: 900 894 \begin{equation} \label{eq:C_1.1} 901 895 \int_D {q\,\;{\textbf{k}}\cdot \frac{1} {e_3} \nabla \times \left( {e_3 \, q \;{\textbf{k}}\times {\textbf{U}}_h } \right)\;dv} \equiv 0 902 896 \end{equation} 903 where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. Indeed, using904 \autoref{eq:dynvor_ens}, the discrete form of the right hand side of \autoref{eq:C_1.1} 905 can be transformed as follow:897 where $dv=e_1\,e_2\,e_3 \; di\,dj\,dk$ is the volume element. 898 Indeed, using \autoref{eq:dynvor_ens}, 899 the discrete form of the right hand side of \autoref{eq:C_1.1} can be transformed as follow: 906 900 \begin{flalign*} 907 901 &\int_D q \,\; \textbf{k} \cdot \frac{1} {e_3 } \nabla \times … … 955 949 \end{aligned} } \right. 956 950 \end{equation} 957 where the indices $i_p$ and $k_p$ take the following value :951 where the indices $i_p$ and $k_p$ take the following values: 958 952 $i_p = -1/2$ or $1/2$ and $j_p = -1/2$ or $1/2$, 959 953 and the vorticity triads, ${^i_j}\mathbb{Q}^{i_p}_{j_p}$, defined at $T$-point, are given by: … … 966 960 This formulation does conserve the potential enstrophy for a horizontally non-divergent flow ($i.e.$ $\chi=0$). 967 961 968 Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, 969 similar manipulation can be done for the 3 others. The discrete form of the right hand 970 side of \autoref{eq:C_1.1} applied to this triad only can be transformed as follow: 962 Let consider one of the vorticity triad, for example ${^{i}_j}\mathbb{Q}^{+1/2}_{+1/2} $, 963 similar manipulation can be done for the 3 others. 964 The discrete form of the right hand side of \autoref{eq:C_1.1} applied to 965 this triad only can be transformed as follow: 971 966 972 967 \begin{flalign*} … … 1020 1015 1021 1016 1022 All the numerical schemes used in NEMO are written such that the tracer content 1023 is conserved by the internal dynamics and physics (equations in flux form). 1024 For advection, only the CEN2 scheme ($i.e.$ $2^{nd}$ order finite different scheme) 1025 conserves the global variance of tracer. Nevertheless the other schemes ensure 1026 that the global variance decreases ($i.e.$ they are at least slightly diffusive). 1027 For diffusion, all the schemes ensure the decrease of the total tracer variance, 1028 except the iso-neutral operator. There is generally no strict conservation of mass, 1029 as the equation of state is non linear with respect to $T$ and $S$. In practice, 1030 the mass is conserved to a very high accuracy. 1017 All the numerical schemes used in NEMO are written such that the tracer content is conserved by 1018 the internal dynamics and physics (equations in flux form). 1019 For advection, 1020 only the CEN2 scheme ($i.e.$ $2^{nd}$ order finite different scheme) conserves the global variance of tracer. 1021 Nevertheless the other schemes ensure that the global variance decreases 1022 ($i.e.$ they are at least slightly diffusive). 1023 For diffusion, all the schemes ensure the decrease of the total tracer variance, except the iso-neutral operator. 1024 There is generally no strict conservation of mass, 1025 as the equation of state is non linear with respect to $T$ and $S$. 1026 In practice, the mass is conserved to a very high accuracy. 1031 1027 % ------------------------------------------------------------------------------------------------------------- 1032 1028 % Advection Term … … 1049 1045 1050 1046 1051 Whatever the advection scheme considered it conserves of the tracer content as all 1052 the scheme are written in flux form. Indeed, let $T$ be the tracer and $\tau_u$, $\tau_v$, 1053 and $\tau_w$ its interpolated values at velocity point (whatever the interpolation is), 1047 Whatever the advection scheme considered it conserves of the tracer content as 1048 all the scheme are written in flux form. 1049 Indeed, let $T$ be the tracer and its $\tau_u$, $\tau_v$, and $\tau_w$ interpolated values at velocity point 1050 (whatever the interpolation is), 1054 1051 the conservation of the tracer content due to the advection tendency is obtained as follows: 1055 1052 \begin{flalign*} … … 1067 1064 \end{flalign*} 1068 1065 1069 The conservation of the variance of tracer due to the advection tendency 1070 can be achieved only with the CEN2 scheme, $i.e.$ when 1071 $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$. 1066 The conservation of the variance of tracer due to the advection tendency can be achieved only with the CEN2 scheme, 1067 $i.e.$ when $\tau_u= \overline T^{\,i+1/2}$, $\tau_v= \overline T^{\,j+1/2}$, and $\tau_w= \overline T^{\,k+1/2}$. 1072 1068 It can be demonstarted as follows: 1073 1069 \begin{flalign*} … … 1103 1099 1104 1100 1105 The discrete formulation of the horizontal diffusion of momentum ensures the 1106 conservation of potential vorticity and the horizontal divergence, and the 1107 dissipation of the square of these quantities ($i.e.$ enstrophy and the 1108 variance of the horizontal divergence) as well as the dissipation of the 1109 horizontal kinetic energy. In particular, when the eddy coefficients are 1110 horizontally uniform, it ensures a complete separation of vorticity and 1111 horizontal divergence fields, so that diffusion (dissipation) of vorticity 1112 (enstrophy) does not generate horizontal divergence (variance of the 1113 horizontal divergence) and \textit{vice versa}. 1114 1115 These properties of the horizontal diffusion operator are a direct consequence 1116 of properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}. 1117 When the vertical curl of the horizontal diffusion of momentum (discrete sense) 1118 is taken, the term associated with the horizontal gradient of the divergence is 1119 locally zero. 1101 The discrete formulation of the horizontal diffusion of momentum ensures 1102 the conservation of potential vorticity and the horizontal divergence, 1103 and the dissipation of the square of these quantities 1104 ($i.e.$ enstrophy and the variance of the horizontal divergence) as well as 1105 the dissipation of the horizontal kinetic energy. 1106 In particular, when the eddy coefficients are horizontally uniform, 1107 it ensures a complete separation of vorticity and horizontal divergence fields, 1108 so that diffusion (dissipation) of vorticity (enstrophy) does not generate horizontal divergence 1109 (variance of the horizontal divergence) and \textit{vice versa}. 1110 1111 These properties of the horizontal diffusion operator are a direct consequence of 1112 properties \autoref{eq:DOM_curl_grad} and \autoref{eq:DOM_div_curl}. 1113 When the vertical curl of the horizontal diffusion of momentum (discrete sense) is taken, 1114 the term associated with the horizontal gradient of the divergence is locally zero. 1120 1115 1121 1116 % ------------------------------------------------------------------------------------------------------------- … … 1125 1120 \label{subsec:C.6.1} 1126 1121 1127 The lateral momentum diffusion term conserves the potential vorticity 1122 The lateral momentum diffusion term conserves the potential vorticity: 1128 1123 \begin{flalign*} 1129 1124 &\int \limits_D \frac{1} {e_3 } \textbf{k} \cdot \nabla \times … … 1211 1206 \label{subsec:C.6.3} 1212 1207 1213 The lateral momentum diffusion term dissipates the enstrophy when the eddy 1214 coefficients are horizontally uniform: 1208 The lateral momentum diffusion term dissipates the enstrophy when the eddy coefficients are horizontally uniform: 1215 1209 \begin{flalign*} 1216 1210 &\int\limits_D \zeta \; \textbf{k} \cdot \nabla \times … … 1236 1230 \label{subsec:C.6.4} 1237 1231 1238 When the horizontal divergence of the horizontal diffusion of momentum 1239 (discrete sense) is taken, the term associated with the vertical curl of the 1240 vorticity is zero locally, due to \autoref{eq:DOM_div_curl}. 1241 The resulting term conserves the $\chi$ and dissipates $\chi^2$ 1242 when the eddy coefficients are horizontally uniform. 1232 When the horizontal divergence of the horizontal diffusion of momentum (discrete sense) is taken, 1233 the term associated with the vertical curl of the vorticity is zero locally, due to \autoref{eq:DOM_div_curl}. 1234 The resulting term conserves the $\chi$ and dissipates $\chi^2$ when the eddy coefficients are horizontally uniform. 1243 1235 \begin{flalign*} 1244 1236 & \int\limits_D \nabla_h \cdot … … 1291 1283 \label{sec:C.7} 1292 1284 1293 As for the lateral momentum physics, the continuous form of the vertical diffusion1294 of momentum satisfies several integral constraints. The first two are associated 1295 with the conservation of momentum and the dissipation of horizontal kinetic energy:1285 As for the lateral momentum physics, 1286 the continuous form of the vertical diffusion of momentum satisfies several integral constraints. 1287 The first two are associated with the conservation of momentum and the dissipation of horizontal kinetic energy: 1296 1288 \begin{align*} 1297 1289 \int\limits_D \frac{1} {e_3 }\; \frac{\partial } {\partial k} … … 1306 1298 \end{align*} 1307 1299 1308 The first property is obvious. The second results from: 1300 The first property is obvious. 1301 The second results from: 1309 1302 \begin{flalign*} 1310 1303 \int\limits_D … … 1326 1319 \end{flalign*} 1327 1320 1328 The vorticity is also conserved. Indeed: 1321 The vorticity is also conserved. 1322 Indeed: 1329 1323 \begin{flalign*} 1330 1324 \int \limits_D … … 1346 1340 \end{flalign*} 1347 1341 1348 If the vertical diffusion coefficient is uniform over the whole domain, the 1349 enstrophy is dissipated, $i.e.$ 1342 If the vertical diffusion coefficient is uniform over the whole domain, the enstrophy is dissipated, $i.e.$ 1350 1343 \begin{flalign*} 1351 1344 \int\limits_D \zeta \, \textbf{k} \cdot \nabla \times … … 1378 1371 &\left[ \frac{A_u^{\,vm}} {e_{3uw}} \delta_{k+1/2} \left[ \delta_{j+1/2} \left[ e_{1u}\,u \right] \right] \right] \biggr\} &&\\ 1379 1372 \end{flalign*} 1380 Using the fact that the vertical diffusion coefficients are uniform, and that in 1381 $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so 1382 that: $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, 1383 it follows: 1373 Using the fact that the vertical diffusion coefficients are uniform, 1374 and that in $z$-coordinate, the vertical scale factors do not depend on $i$ and $j$ so that: 1375 $e_{3f} =e_{3u} =e_{3v} =e_{3t} $ and $e_{3w} =e_{3uw} =e_{3vw} $, it follows: 1384 1376 \begin{flalign*} 1385 1377 \equiv A^{\,vm} \sum\limits_{i,j,k} \zeta \;\delta_k … … 1398 1390 \left( \frac{A^{\,vm}} {e_3 }\; \frac{\partial \textbf{U}_h } {\partial k} \right) \right)\; dv = 0 &&&\\ 1399 1391 \end{flalign*} 1400 and the square of the horizontal divergence decreases ($i.e.$ the horizontal 1401 divergence is dissipated) if the vertical diffusion coefficient is uniform over the 1402 whole domain: 1392 and the square of the horizontal divergence decreases ($i.e.$ the horizontal divergence is dissipated) if 1393 the vertical diffusion coefficient is uniform over the whole domain: 1403 1394 1404 1395 \begin{flalign*} … … 1463 1454 \label{sec:C.8} 1464 1455 1465 The numerical schemes used for tracer subgridscale physics are written such 1466 th at the heat and salt contents are conserved (equations in flux form).1467 Since a flux form is used to compute the temperature and salinity, 1468 the quadratic form of these quantities ($i.e.$ their variance) globally tends to diminish. 1456 The numerical schemes used for tracer subgridscale physics are written such that 1457 the heat and salt contents are conserved (equations in flux form). 1458 Since a flux form is used to compute the temperature and salinity, 1459 the quadratic form of these quantities ($i.e.$ their variance) globally tends to diminish. 1469 1460 As for the advection term, there is conservation of mass only if the Equation Of Seawater is linear. 1470 1461
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