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lecture6:primequation [2015/03/17 09:03] – created - external edit 127.0.0.1lecture6:primequation [2025/03/26 11:17] (current) – [Transport Equations for Tracer] matt
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 \begin{equation*} \begin{equation*}
 +
 \vec{J} = -D {\nabla}C \vec{J} = -D {\nabla}C
 +
 \end{equation*} \end{equation*}
  
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   * The mathematical term for the accumulation rate is     * The mathematical term for the accumulation rate is  
-\begin{equation*}(\frac{\partial{A}}{\partial{t}}}) {\partial x}{\partial y}{\partial z}\end{equation*} + 
 +\begin{equation*} 
 +\frac{\partial A}{\partial t} {\partial x}{\partial y}{\partial z} 
 +\end{equation*} 
  
   * The mathematical term for the flux rate of A into the volume is a bit more complicated. The flux's, or flow through an area's, approximation is derived for each component at a time, which will give us    * The mathematical term for the flux rate of A into the volume is a bit more complicated. The flux's, or flow through an area's, approximation is derived for each component at a time, which will give us 
 +
 +
 \begin{equation*} \begin{equation*}
--(\frac{\partial}{\partial x}}(Au) + + 
-\frac{\partial}{\partial y}}(Av) + +-\left(\frac{\partial}{\partial x}(Au) + 
-\frac{\partial}{\partial z}}(Aw))+\frac{\partial}{\partial y}(Av) + 
 +\frac{\partial}{\partial z}(Aw)\right)
 {\partial x}{\partial y}{\partial z} {\partial x}{\partial y}{\partial z}
 +
 \end{equation*} \end{equation*}
  
-  * The mathematical term for the generation rate is similar to the flux rate, but will also account for changes in time. It is  \begin{equation*} +  * The mathematical term for the generation rate is similar to the flux rate, but will also account for changes in time. It is   
-(\frac{\partial {A}}{\partial{t}}} + + 
-\frac{\partial}{\partial x}}(Au) + +\begin{equation*} 
-\frac{\partial}{\partial y}}(Av) + +\left(\frac{\partial A}{\partial t} + 
-\frac{\partial}{\partial z}}(Aw))+\frac{\partial}{\partial x}(Au) + 
 +\frac{\partial}{\partial y}(Av) + 
 +\frac{\partial}{\partial z}(Aw) \right)
 {\partial x}{\partial y}{\partial z} {\partial x}{\partial y}{\partial z}
 \end{equation*} \end{equation*}
  
 and for a randomly small parcel volume this will result in and for a randomly small parcel volume this will result in
 +
 \begin{equation*} \begin{equation*}
-\frac{\partial {A}}{\partial{t}}} + +\frac{\partial {A}}{\partial{t}} + 
-\frac{\partial}{\partial x}}(Au) + +\frac{\partial}{\partial x}(Au) + 
-\frac{\partial}{\partial y}}(Av) + +\frac{\partial}{\partial y}(Av) + 
-\frac{\partial}{\partial z}}(Aw)+\frac{\partial}{\partial z}(Aw)
 = S(A), = S(A),
 \end{equation*} \end{equation*}
 +
 where S(A) describes the net generation, which accounts for both sources and drains.\\  where S(A) describes the net generation, which accounts for both sources and drains.\\ 
  
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 \begin{equation*} \begin{equation*}
-\frac{\partial {\rho}}{\partial{t}}} + +\frac{\partial \rho}{\partial t} + 
-\frac{\partial}{\partial x}}(\rho u) + +\frac{\partial}{\partial x}(\rho u) + 
-\frac{\partial}{\partial y}}(\rho v) + +\frac{\partial}{\partial y}(\rho v) + 
-\frac{\partial}{\partial z}}(\rho w)+\frac{\partial}{\partial z}(\rho w)
 = 0, = 0,
 \end{equation*} \end{equation*}
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 === Term definitions === === Term definitions ===
 ^ Definition      ^ Term     ^ ^ Definition      ^ Term     ^
-|Material Derivative Operator |$  +|Material Derivative Operator | $ \frac{D}{Dt} := \frac{\partial}{\partial t} + u \frac{\partial}{\partial x} + v \frac{\partial}{\partial y} + w \frac{\partial}{\partial z} =  \frac{\partial}{\partial t}+\vec{u} \cdot \nabla $ |
-\frac{D}{D{t}} :=  +
-\frac{\partial}{\partial {t}} + +
-u \frac{\partial}{\partial{x}} +  +
-v \frac{\partial}{\partial{y}} + +
-w \frac{\partial}{\partial{z}} =  +
-\frac{\partial}{\partial{t}}+ +
-\vec{u} \cdot \nabla  +
-$|+
 |Local Derivative|$ \frac{\partial }{\partial t}$| |Local Derivative|$ \frac{\partial }{\partial t}$|
 |Advection|$ \vec{u} \cdot \nabla\rho $| |Advection|$ \vec{u} \cdot \nabla\rho $|
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 \begin{equation*} \begin{equation*}
-\(S (C_i)= Q_i &\neq\ 0 + S (C_i)= Q_i \neq\ 0 
 \end{equation*}   \end{equation*}  
  
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 \begin{equation*} \begin{equation*}
-\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) 
-= Q_i,+= Q_i
 \end{equation*} \end{equation*}
  
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 \begin{equation*} \begin{equation*}
-\frac{D{C_i}}{D{t}} + C_i\cdot  +\frac{DC_i}{Dt} + C_i\cdot  
-\left( \frac{\partial{u}}{\partial{x}} ++( \frac{\partial u}{\partial x} +
 \frac{\partial{v}}{\partial{y}} + \frac{\partial{v}}{\partial{y}} +
-\frac{\partial{w}}{\partial{z}} \right)  = Q_i+\frac{\partial{w}}{\partial{z}} )  = Q_i
 \end{equation*} \end{equation*}
  
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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} )  +|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} $|
-$|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} $|+
  
 ==== Temperature Equation ==== ==== Temperature Equation ====
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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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