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lecture6:primequation [2025/03/26 11:12] – [The Principle of Mass Conservation] mattlecture6:primequation [2025/03/26 11:17] (current) – [Transport Equations for Tracer] matt
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 \frac{\partial C_i}{\partial t} + \frac{\partial C_i}{\partial t} +
 \frac{\partial}{\partial x}(C_i u) + \frac{\partial}{\partial x}(C_i u) +
- 
 \frac{\partial}{\partial y}(C_i v) + \frac{\partial}{\partial y}(C_i v) +
 \frac{\partial}{\partial z}(C_i w) \frac{\partial}{\partial z}(C_i w)
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 ^ Definition      ^ Equation     ^ Term definition      ^ Term     ^ ^ Definition      ^ Equation     ^ Term definition      ^ Term     ^
-|Tracer Equation (no diffusion) |$ +|Tracer Equation (no diffusion) |$ Q_i=\frac{\partial{C_i}}{\partial{t}} + \nabla  \cdot (C_i \vec{u} ) $ |Change of parcel concentration \\ Local Derivative \\ \\ Sink/Source|$ \frac{\partial{C_i}}{\partial{t}} $ \\ \\ \\ $ Q_i $| 
-Q_i=\frac{\partial{C_i}}{\partial{t}} +  +|Tracer Equation with diffusion ((This includes Fick's first law: \begin{equation*}\vec{J_D} = - k_i \vec{\nabla} C_i\end{equation*} ))|$Q_i  -\nabla \cdot \vec{J_D}   = \frac{\partial{C_i}}{\partial{t}} + \nabla  \cdot (C_i \vec{u} ) $|Diffusion |$ -\nabla \cdot \vec{J_D} $|
-\nabla  \cdot (C_i \vec{u} )  +
-$|Change of parcel concentration \\ Local Derivative \\ \\ Sink/Source|$ \frac{\partial{C_i}}{\partial{t}} $ \\ \\ \\ $ Q_i $| +
-|Tracer Equation with diffusion ((This includes Fick's first law: \begin{equation*}\vec{J_D} = - k_i \vec{\nabla} C_i\end{equation*} ))|$ +
-Q_i  -\nabla \cdot \vec{J_D}   = +
- \frac{\partial{C_i}}{\partial{t}} +  +
-\nabla  \cdot (C_i \vec{u} )  +
-$|Diffusion |$ -\nabla \cdot \vec{J_D} $|+
 |Tracer Equation for Seasalt S \\ (for $ C_{i} = S$ and no salt sources  $ Q_{S} = 0$)|$ \rho \frac{DS}{Dt} = \nabla \cdot (\rho{k_s} \nabla {S}) $|Material Derivative of Salinity |$ \frac{DS}{Dt} $| |Tracer Equation for Seasalt S \\ (for $ C_{i} = S$ and no salt sources  $ Q_{S} = 0$)|$ \rho \frac{DS}{Dt} = \nabla \cdot (\rho{k_s} \nabla {S}) $|Material Derivative of Salinity |$ \frac{DS}{Dt} $|
  
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 \end{equation*} \end{equation*}
  
-To receive the final version of this equation it needs to account for sources and sinks, making $S(A)&\neq 0$. Such sources and sinks could be an internal heat source $Q_T$ or an external heat flux $\vec{J_T}$, e.g., solar radiation.+To receive the final version of this equation it needs to account for sources and sinks, making $S(A)\neq 0$. Such sources and sinks could be an internal heat source $Q_T$ or an external heat flux $\vec{J_T}$, e.g., solar radiation.
  
 \begin{equation*} \begin{equation*}

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