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lecture6:primequation [2025/03/26 09:47] – [Transport Equations for Tracer] mattlecture6:primequation [2025/03/26 11:17] (current) – [Transport Equations for Tracer] matt
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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 
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 \begin{equation*} \begin{equation*}
  
--(\frac{\partial}{\partial x}}(Au) + +-\left(\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)\right)
 {\partial x}{\partial y}{\partial z} {\partial x}{\partial y}{\partial z}
  
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 \begin{equation*} \begin{equation*}
-(\frac{\partial {A}}{\partial{t}}} + +\left(\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) \right)
 {\partial x}{\partial y}{\partial z} {\partial x}{\partial y}{\partial z}
 \end{equation*} \end{equation*}
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 \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*}
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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  +
-$|+
 |Local Derivative|∂∂t| |Local Derivative|∂∂t|
 |Advection|→u⋅∇ρ| |Advection|→u⋅∇ρ|
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 \frac{\partial C_i}{\partial t} + \frac{\partial C_i}{\partial t} +
 \frac{\partial}{\partial x}(C_i u) + \frac{\partial}{\partial x}(C_i u) +
- 
 \frac{\partial}{\partial y}(C_i v) + \frac{\partial}{\partial y}(C_i v) +
 \frac{\partial}{\partial z}(C_i w) \frac{\partial}{\partial z}(C_i w)
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 ^ Definition      ^ Equation     ^ Term definition      ^ Term     ^ ^ Definition      ^ Equation     ^ Term definition      ^ Term     ^
-|Tracer Equation (no diffusion) |$ +|Tracer Equation (no diffusion) |Qi=∂Ci∂t+∇⋅(Ci→u) |Change of parcel concentration \\ Local Derivative \\ \\ Sink/Source|∂Ci∂t \\ \\ \\ Qi| 
-Q_i=\frac{\partial{C_i}}{\partial{t}} +  +|Tracer Equation with diffusion ((This includes Fick's first law: →JD=−ki→∇Ci ))|Qi−∇⋅→JD=∂Ci∂t+∇⋅(Ci→u)|Diffusion |−∇⋅→JD|
-\nabla  \cdot (C_i \vec{u} )  +
-|ChangeofparcelconcentrationLocalDerivativeSink/Source| \frac{\partial{C_i}}{\partial{t}} Q_i $| +
-|Tracer Equation with diffusion ((This includes Fick's first law: →JD=−ki→∇Ci ))|$ +
-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 Ci=S and no salt sources  QS=0)|ρDSDt=∇⋅(ρks∇S)|Material Derivative of Salinity |DSDt| |Tracer Equation for Seasalt S \\ (for Ci=S and no salt sources  QS=0)|ρDSDt=∇⋅(ρks∇S)|Material Derivative of Salinity |DSDt|
  
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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 $S(A)&\neq 0.SuchsourcesandsinkscouldbeaninternalheatsourceQ_Toranexternalheatflux\vec{J_T}$, e.g., solar radiation.+To receive the final version of this equation it needs to account for sources and sinks, making S(A)≠0. Such sources and sinks could be an internal heat source QT or an external heat flux →JT, e.g., solar radiation.
  
 \begin{equation*} \begin{equation*}

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