Changes between Version 3 and Version 4 of u/erica/d

Timestamp:

12/05/15 14:13:05 (9 years ago)

Author:

Erica Kaminski

Comment:

—

u/erica/d

@@ -12 +12 @@
 [[latex($(\bold{\nabla} \times \bold{B})\times \bold{B}=-\bold{\nabla}(\frac{B^2}{2 \mu}) + \frac{1}{\mu}(\bold{B}\cdot \bold{\nabla}) \bold{B}$)]]
 
-the first term on the RHS is the same form as thermal pressure and so we call it the magnetic pressure. The second term is generally called the magnetic tension. To see why this is so, we rewrite it as:
+the first term on the RHS is the same form as thermal pressure and so we call it the magnetic pressure. The second term is generally called the magnetic tension. It is the directional derivative of B along B, or in other words, disappears when B is straight and uncurved. Thus, it acts as a restoring force to straighten B out. It can be rewritten as:
 
 [[latex($\frac{1}{\mu}(\bold{B}\cdot \bold{\nabla}) \bold{B} = \hat{\bold{b}}\hat{\bold{b}} \cdot \bold{\nabla}(\frac{B^2}{2 \mu}) + \frac{B^2}{\mu} \frac{\hat{\bold{n}}}{R_c}$)]] 
@@ -28 +28 @@
 
 [[latex($\hat{\bold{b}}\hat{\bold{b}} \cdot \bold{\nabla}(\frac{B^2}{2 \mu}) = \bold{0}$)]]
+
+Under this situation the momentum equation is:
+
+[[latex($\rho \frac{d}{dt} \bold{u} + \rho (\bold{u} \cdot \bold{\nabla}) \bold{u} = - \bold{\nabla} P -\bold{\nabla}(\frac{B^2}{2 \mu}) + \frac{B^2}{\mu} \frac{\hat{\bold{n}}}{R_c} $)]]
+
+leaving us with the magnetic pressure term discussed above, and a term that points perpendicularly to the field line toward the center of the curvature (remember the tension term as a whole only arises when the field line is bent). Here [[latex($R_c$)]] is the radius of curvature.