Radially weighted Backus- and Childress-type bounds for spherical dynamos

In MHD dynamo theory well-known necessary criteria for dynamo action are formulated in terms of lower bounds either on the maximum modulus of the velocity field (Childress-type) or the maximum strain of the velocity field (Backus-type). We generalize these criteria for spherical dynamos by introducing a radially varying weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis r right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">f(r)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The corresponding <italic>l</italic> ower <italic>b</italic> ound Reynolds numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript l b Superscript upper C Baseline left-bracket f right-bracket"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>l</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>C</mml:mi> </mml:msubsup> <mml:mo stretchy="false">[</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R_{lb}^C [f]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (based on velocity) and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript l b Superscript upper B Baseline left-bracket f right-bracket"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>l</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> <mml:mi>B</mml:mi> </mml:msubsup> <mml:mo stretchy="false">[</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">R_{lb}^B [f]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (based on strain) are determined for two types of such weights: a power law profile <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis r right-parenthesis equals r Superscript alpha"> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mi> α </mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">f(r) = r^\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to alpha less-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi> α </mml:mi> <mml:mo> ≤ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">0\leq \alpha \leq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and an optimal radial profile <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Subscript bold v"> <mml:semantics> <mml:msub> <mml:mi>f</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">v</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">f_\mathbf {v}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> depending on the velocity field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold v"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">v</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {v}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in question. To assess the quality of these lower bounds we compare them with weighted critical Reynolds numbers <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript c Superscript upper C"> <mml:semantics> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> <mml:mi>C</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">R_c^C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (Childress-type) and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R Subscript c Superscript upper B"> <mml:semantics> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mi>c</mml:mi> <mml:mi>B</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">R_c^B</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (Backus-type), respectively, for the onset of dynamo action of the well-known efficient <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="s 2 t 2"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>t</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">s_2t_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> velocity field [Dudley and James, <italic>Time-dependent kinematic dynamos with stationary flows</italic> , Proc. Roy. Soc. London Ser. A 425 (1989), no. 1869, 407–429] and