22.615, MHD Theory of Fusion Systems Prof. Freidberg Lecture 7: The First Order Grad Shafranov Equation. dp 1 dp

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1 First Order Eqution.65, MHD Theory of Fusion Systems Prof. Freiderg Lecture 7: The First Order Grd Shfrnov Eqution The first order Grd Shfrnov eqution is given y d p d dp d + μr + R B B = μr r cos + cos d d d R dr Simplify This eqution is simplified s follows: d R dr. B =, = ( r) dp R dp dp. R = = d dr B dr 3. μ dp RHS = B r cos B dr d d μ RB dr RB dr 4. = R ( p + B B ) d B d d d = rb rb = B dr B r dr B dr r dr 5. d d μr dp rb B cos B dr = + r dr B dr Solve This eqution is solved s follows:. Note tht ll the forcing terms re proportionl to cos.. The oundry conditions for circle of rdius re given y (, ) const = ( ) (, ) = + (, ) =.65, MHD Theory of Fusion Systems Lecture 7 Prof. Freiderg Pge of 7

2 . For n ellipse r = + δcos. Assume δ. + δcos, + δcos +, +..., = δcos second hrmonic is required in the solution 3. Thus, for circulr oundry with cos driving terms we cn write ( r, ) = + cos explicit cos dependence ( ) = 4. Simplify the eqution B B rp B + = B r B r μ r r B rb μr = B r rb B p 5. Note B rb = rb r B ( ) B r rb rb r B = + 6. d d rb = rb μr dp dr dr B dr 7. d r dp = yb y dy dr B μ rb dy regulrity t r = 8. dx dp = B B y dy dy x y r μ xb 9. This expression for represents the toroidl correction to the equilirium solution..65, MHD Theory of Fusion Systems Lecture 7 Prof. Freiderg Pge of 7

3 Consequences of Toroidicity. The min consequence is n outwrd shift of the flux surfces.. From it is strightforwrd to clculte 3.. ( r, ) = + cos = const. the flux surfce shift.. Assume the flux surfces re of the form r r r c. Then ( r ) r = const. cos = +., d. Solve for r ( ) cos r r, = = r cos 4. The eqution for the flux surfces is given y r r ssuming r = + cos, 5.. Note tht x + y =r is the eqution of shifted circle, equivlent to the eqution for r.. Let x = r cos y = r sin cos r r r r cos r r r r + cos 6. The flux surfces re shifted circles, with shift =.65, MHD Theory of Fusion Systems Lecture 7 Prof. Freiderg Pge 3 of 7

4 The Shfrnov Shift. Clculte the Shfrnov shift ( ) crry current, i.e., the edge of the plsm. where is the lst surfce to. Simplify given y dx x dp ( ) = B ( ) yb μ y dy dy xb 3. Consider the term T, noting tht < x <. μ x y pdy = μ y pdy y p = μ + 4 μ py dy. T B 3 ( ) I μ = 4π β p ( ) = B β dx = β p xb c. For > = x B B x p d. Therefore.65, MHD Theory of Fusion Systems Lecture 7 Prof. Freiderg Pge 4 of 7

5 = p T B β xdx β B p = 4. Consider the term T. Seprte the integrl into two prts dx x = ( ) T B xb yb dy dx x = B ( ) yb dy + yb dy xb : dy T yb dy = B = B l nx ln y x x. c. Sustitute T = B x ln x ln ln + ln B = ln = B ( ) ln 4 d. T = B ( ) yb xb dx dy e. Introduce the normlized internl inductnce per unir length l i. f. This follows from B LI i = dr p μ = ( πr )( π) B r dr μ π R μ = Brdr.65, MHD Theory of Fusion Systems Lecture 7 Prof. Freiderg Pge 5 of 7

6 L g. Define l i μ i π R 4 π to μ 4π the internl inductnce per unit length normlized h. Then = π i R 4π Brdr ( ) μ μ πb ( Rl) π μ B ( l ) i = i. l i is profile prmeter relted to the width of the J φ profile j. Then li T = B ( ) x d x l i = B ( ) 4 5. Comine these terms to evlute. ( ) ( ) = = RB ( ) ( ). Then B p l β i = + B ( ) ln B ( ) RB ( ) 4 4 c. The Shfrnov shift is given y = + + ln R li βp Properties of the Shfrnov Shift.. The shift is smll, implying tht our pproximtions re R consistent..65, MHD Theory of Fusion Systems Lecture 7 Prof. Freiderg Pge 6 of 7

7 . 3. ( ) βp. This is the outwrd shift due to the tire tue force nd the /R force. l i + + ln internl field externl field hoop force This is the shift due to the hoop force..65, MHD Theory of Fusion Systems Lecture 7 Prof. Freiderg Pge 7 of 7