|
33 | 33 | "\n", |
34 | 34 | "The general form for the salt/salinity budget can be formulated in the same way as with the [heat budget](https://ecco-v4-python-tutorial.readthedocs.io/ECCO_v4_Heat_budget_closure.html#Introduction), where instead of potential temperature ($\\theta$), the budget is described with salinity ($S$).\n", |
35 | 35 | "\n", |
36 | | - "\\begin{equation}\\label{general}\n", |
37 | | - "\\begin{aligned}\n", |
| 36 | + "$$\n", |
38 | 37 | "\\frac{\\partial S}{\\partial t} = -\\nabla \\cdot (S \\mathbf{u}) - \\nabla\\cdot\\mathbf{F}_\\textrm{diff}^{S} + {F}_\\textrm{forc}^{S}\n", |
39 | | - "\\end{aligned}\n", |
40 | | - "\\end{equation}\n", |
| 38 | + "$$\n", |
41 | 39 | "\n", |
42 | 40 | "The total tendency ($\\frac{\\partial S}{\\partial t}$) is equal to advective convergence ($-\\nabla \\cdot (S \\mathbf{u})$), diffusive flux convergence ($-\\nabla \\cdot \\mathbf{F}_\\textrm{diff}^{S}$) and a forcing term ${F}_\\textrm{forc}^{S}$." |
43 | 41 | ] |
|
48 | 46 | "source": [ |
49 | 47 | "In the case of ECCOv4, salt is strictly a conserved mass and can be described as\n", |
50 | 48 | "\n", |
51 | | - "\\begin{equation}\\label{slt}\n", |
| 49 | + "$$\n", |
52 | 50 | "\\underbrace{\\frac{\\partial(s^* S)}{\\partial t}}_{G^{Slt}_\\textrm{total}} = \\underbrace{-\\nabla_{z^{*}} \\cdot(s^* S \\, \\mathbf{v}_{res}) - \\frac{\\partial(S\\,w_{res})}{\\partial z^* }}_{G^{Slt}_\\textrm{advection}}\\underbrace{- s^* ({\\nabla\\cdot\\mathbf{F}_\\textrm{diff}^{S}})}_{G^{Slt}_\\textrm{diffusion}} + \\underbrace{s^* {F}_\\textrm{forc}^{S}}_{G^{Slt}_\\textrm{forcing}}\n", |
53 | | - "\\end{equation}\n", |
| 51 | + "$$\n", |
54 | 52 | "\n", |
55 | 53 | "The change in salt content over time ($G^{Slt}_\\textrm{total}$) is equal to the convergence of the advective flux ($G^{Slt}_\\textrm{advection}$) and diffusive flux ($G^{Slt}_\\textrm{diffusion}$) plus a forcing term associated with surface salt exchanges ($G^{Slt}_\\textrm{forcing}$). As with the [heat budget](https://ecco-v4-python-tutorial.readthedocs.io/ECCO_v4_Heat_budget_closure.html#Introduction), we present both the horizontal ($\\mathbf{v}_{res}$) and vertical ($w_{res}$) components of the advective term. Again, we have $\\mathbf{u}_{res}$ as the \"residual mean\" velocities, which contain both the resolved (Eulerian) and parameterizing \"GM bolus\" velocities. Also note the use of the rescaled height coordinate $z^*$ and the scale factor $s^*$ which have been described in the [volume](https://ecco-v4-python-tutorial.readthedocs.io/ECCO_v4_Volume_budget_closure.html#ECCOv4-Global-Volume-Budget-Closure) and [heat](https://ecco-v4-python-tutorial.readthedocs.io/ECCO_v4_Heat_budget_closure.html#ECCOv4-Global-Heat-Budget-Closure) budget tutorials." |
56 | 54 | ] |
|
61 | 59 | "source": [ |
62 | 60 | "The salt budget in ECCOv4 only considers the mass of salt in the ocean. Thus, the convergence of freshwater and surface freshwater exchanges are not formulated specifically. An important point here is that, given the nonlinear free surface condition in ECCOv4, budgets for salt content (an extensive quantity) are not the same as budgets for salinity (an intensive quantity). In order to accurately describe variation in salinity, we need to take into account the variation of both salt and volume. Using the product rule, $G^{Slt}_\\textrm{total}$ (i.e., the left side of the salt budget equation) can be extended as follows\n", |
63 | 61 | "\n", |
64 | | - "\\begin{equation}\\label{s_star}\n", |
| 62 | + "$$\n", |
65 | 63 | "\\frac{\\partial(s^* S)}{\\partial t} = s^* \\frac{\\partial S}{\\partial t} + S \\frac{ \\partial s^* }{\\partial t}\n", |
66 | | - "\\end{equation}" |
| 64 | + "$$" |
67 | 65 | ] |
68 | 66 | }, |
69 | 67 | { |
|
80 | 78 | "metadata": {}, |
81 | 79 | "source": [ |
82 | 80 | "Since $s^* = 1 + \\frac{\\eta}{H}$ we can define the temporal change in $s^*$ as\n", |
83 | | - "\\begin{equation}\\label{s_star_dt}\n", |
| 81 | + "$$\n", |
84 | 82 | "\\frac{\\partial s^*}{\\partial t} = \\frac{1}{H}\\,\\frac{\\partial \\eta}{\\partial t}\n", |
85 | | - "\\end{equation}\n", |
| 83 | + "$$\n", |
86 | 84 | "\n", |
87 | 85 | "This constitutes the conservation of volume in ECCOv4, which can be formulated as\n", |
88 | 86 | "\n", |
89 | | - "\\begin{equation}\\label{vol_ecco}\n", |
| 87 | + "$$\n", |
90 | 88 | "\\frac{1}{H}\\,\\frac{\\partial \\eta}{\\partial t} = -\\nabla_{z^* } \\cdot (s^*\\mathbf{v}) - \\frac{\\partial w}{\\partial z^* } + \\mathcal{F}\n", |
91 | | - "\\end{equation}\n", |
| 89 | + "$$\n", |
92 | 90 | "\n", |
93 | 91 | "You can read more about volume conservation and the $z^*$ coordinate system in another [tutorial](https://ecco-v4-python-tutorial.readthedocs.io/ECCO_v4_Volume_budget_closure.html#ECCOv4-Global-Volume-Budget-Closure). $\\mathcal{F}$ denotes the volumetric surface fluxes and can be decomposed into net atmospheric freshwater fluxes (i.e., precipitation minus evaporation, $P - E$), continental runoff ($R$) and exchanges due to sea ice melting/formation ($I$). Here $\\mathbf{v} = (u, v)$ and $w$ are the resolved horizontal and vertical velocities, respectively." |
94 | 92 | ] |
|
99 | 97 | "source": [ |
100 | 98 | "Thus, the conservation of salinity in ECCOv4 can be described as\n", |
101 | 99 | "\n", |
102 | | - "\\begin{equation}\\label{sln}\n", |
| 100 | + "$$\n", |
103 | 101 | "\\underbrace{\\frac{\\partial S}{\\partial t}}_{G^{Sln}_\\textrm{total}} = \\underbrace{\\frac{1}{s^* }\\,\\left[S\\,\\left(\\nabla_{z^* } \\cdot (s^* \\mathbf{v}) + \\frac{\\partial w}{\\partial z^* }\\right) - \\nabla_{z^* } \\cdot (s^* S \\, \\mathbf{v}_{res}) - \\frac{\\partial(S\\,w_{res})}{\\partial z^* }\\right]}_{G^{Sln}_\\textrm{advection}} \\underbrace{- \\nabla \\cdot \\mathbf{F}_\\textrm{diff}^{S}}_{G^{Sln}_\\textrm{diffusion}} + \\underbrace{F_\\textrm{forc}^{S} - S\\,\\mathcal{F}}_{G^{Sln}_\\textrm{forcing}}\n", |
104 | | - "\\end{equation}\n", |
| 102 | + "$$\n", |
105 | 103 | "\n", |
106 | 104 | "Notice here that, in contrast to the salt budget equation, the salinity equation explicitly includes the surface forcing ($S\\,\\mathcal{F}$). $\\mathcal{F}$ represents surface freshwater exchanges ($P-E+R-I$) and $F_{\\textrm{forc}}^{S}$ represents surface salt fluxes (i.e., addition/removal of salt). Besides the convergence of the advective flux ($\\nabla\\cdot(S\\mathbf{u}_{res})$), the salinity equation also includes the convergence of the volume flux multiplied by the salinity ($S\\,(\\nabla\\cdot\\mathbf{u})$), which accounts for the concentration/dilution effect of convergent/divergent volume flux." |
107 | 105 | ] |
|
112 | 110 | "source": [ |
113 | 111 | "The (liquid) freshwater content is defined here as the volume of freshwater (i.e., zero-salinity water) that needs to be added (or subtracted) to account for the deviation between salinity $S$ from a given reference salinity $S_{ref}$. Thus, within a control volume $V$ the freshwater content is defined as a volume ($V_{fw}$):\n", |
114 | 112 | "\n", |
115 | | - "\\begin{equation}\\label{fw}\n", |
| 113 | + "$$\n", |
116 | 114 | "V_{fw} = \\iiint_V\\frac{S_{ref} - S}{S_{ref}}dV\n", |
117 | | - "\\end{equation}\n", |
| 115 | + "$$\n", |
118 | 116 | "\n", |
119 | 117 | "Similar to the salt and salinity budgets, the total tendency (i.e., change in freshwater content over time) can be expressed as the sum of the tendencies due to advective convergence, diffusive convergence, and forcing:\n", |
120 | 118 | "\n", |
121 | | - "\\begin{equation}\n", |
| 119 | + "$$\n", |
122 | 120 | "\\underbrace{\\frac{\\partial V_{fw}}{\\partial t}}_{G^{fw}_\\textrm{total}} = \\underbrace{-\\nabla \\cdot \\mathbf{F}_\\textrm{adv}^{fw}}_{G^{fw}_\\textrm{advection}} \\underbrace{- \\nabla \\cdot \\mathbf{F}_\\textrm{diff}^{fw}}_{G^{fw}_\\textrm{diffusion}} + \\underbrace{\\mathcal{F}}_{G^{fw}_\\textrm{forcing}}\n", |
123 | | - "\\end{equation}\n", |
| 121 | + "$$\n", |
124 | 122 | "\n", |
125 | 123 | "\n", |
126 | 124 | "## Datasets to download\n", |
|
2141 | 2139 | "source": [ |
2142 | 2140 | "Advective fluxes of freshwater are calculated offline using salinity and velocity fields to calculate the volume convergence of freshwater:\n", |
2143 | 2141 | "\n", |
2144 | | - "\\begin{equation}\\label{Fw_adv}\n", |
| 2142 | + "$$\n", |
2145 | 2143 | "\\mathbf{\\mathcal{F}_{adv}} = \\iint_A\\mathbf{u}_{res} \\cdot \\left(\\frac{S_{ref} - S}{S_{ref}}\\right)dA\n", |
2146 | | - "\\end{equation}\n", |
| 2144 | + "$$\n", |
2147 | 2145 | "\n", |
2148 | 2146 | "$u_{res}$ is the residual mean velocity field, which contains both the resolved (Eulerian), as well as the Gent-McWilliams bolus velocity (i.e., the parameterization of unresolved eddy effects)." |
2149 | 2147 | ] |
|
2261 | 2259 | "### Freshwater forcing\n", |
2262 | 2260 | "\n", |
2263 | 2261 | "Include salinity forcing as follows:\n", |
2264 | | - "$ \\frac{\\partial fw}{\\partial t} = -\\frac{1}{S_{ref}}\\frac{\\partial S}{\\partial t} $\n", |
| 2262 | + "$$ \\frac{\\partial fw}{\\partial t} = -\\frac{1}{S_{ref}}\\frac{\\partial S}{\\partial t} $$\n", |
2265 | 2263 | "\n", |
2266 | 2264 | "so $G^{fw}_\\textrm{forcing}= -\\frac{1}{S_{ref}} G^{Sln}_\\textrm{forcing}$" |
2267 | 2265 | ] |
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