Wall modes and the transition to bulk convection in rotating Rayleigh-Bénard convection

Abstract

We investigate states of rapidly rotating Rayleigh-Bénard convection in a cylindrical cell over a range of Rayleigh numbers <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>3</a:mn><a:mo>×</a:mo><a:msup><a:mn>10</a:mn><a:mn>5</a:mn></a:msup><a:mo>≤</a:mo><a:mtext>Ra</a:mtext><a:mo>≤</a:mo><a:mn>5</a:mn><a:mo>×</a:mo><a:msup><a:mn>10</a:mn><a:mn>9</a:mn></a:msup></a:mrow></a:math> and Ekman numbers <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msup><b:mn>10</b:mn><b:mrow><b:mo>−</b:mo><b:mn>6</b:mn></b:mrow></b:msup><b:mo>≤</b:mo><b:mtext>Ek</b:mtext><b:mo>≤</b:mo><b:msup><b:mn>10</b:mn><b:mrow><b:mo>−</b:mo><b:mn>4</b:mn></b:mrow></b:msup></b:mrow></b:math> for Prandtl number <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mtext>Pr</c:mtext><c:mo>=</c:mo><c:mn>0.8</c:mn></c:mrow></c:math> and aspect ratios <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mn>1</d:mn><d:mo>/</d:mo><d:mn>5</d:mn><d:mo>≤</d:mo><d:mi mathvariant="normal">Γ</d:mi><d:mo>≤</d:mo><d:mn>5</d:mn></d:mrow></d:math> using direct numerical simulations. We characterize, for perfectly insulating sidewall boundary conditions, the first transition to convection via wall mode instability and the nonlinear growth and instability of the resulting wall mode states, including a secondary transition to time dependence. We show how the radial structure of the vertical velocity <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:msub><f:mi>u</f:mi><f:mi>z</f:mi></f:msub></f:math> and the temperature <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>T</g:mi></g:math> is captured well by the linear eigenfunctions of the wall mode instability where the radial width of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>u</h:mi><h:mi>z</h:mi></h:msub></h:math> is <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"><i:mrow><i:msub><i:mi>δ</i:mi><i:msub><i:mi>u</i:mi><i:mi>z</i:mi></i:msub></i:msub><i:mo>∼</i:mo><i:msup><i:mtext>Ek</i:mtext><i:mrow><i:mn>1</i:mn><i:mo>/</i:mo><i:mn>3</i:mn></i:mrow></i:msup><i:mi>r</i:mi><i:mo>/</i:mo><i:mi>H</i:mi></i:mrow></i:math> whereas <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"><j:mrow><j:msub><j:mi>δ</j:mi><j:mi>T</j:mi></j:msub><j:mo>∼</j:mo><j:msup><j:mi>e</j:mi><j:mrow><j:mo>−</j:mo><j:mi>k</j:mi><j:mi>r</j:mi></j:mrow></j:msup></j:mrow></j:math> (<k:math xmlns:k="http://www.w3.org/1998/Math/MathML"><k:mi>k</k:mi></k:math> is the wave number of a laterally infinite wall mode state). The disparity in spatial scales for <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"><l:mrow><l:mtext>Ek</l:mtext><l:mo>=</l:mo><l:msup><l:mn>10</l:mn><l:mrow><l:mo>−</l:mo><l:mn>6</l:mn></l:mrow></l:msup></l:mrow></l:math> means that the heat transport is dominated by the radial structure of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>u</m:mi><m:mi>z</m:mi></m:msub></m:math> since <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"><n:mi>T</n:mi></n:math> varies slowly over the radial scale <o:math xmlns:o="http://www.w3.org/1998/Math/MathML"><o:msub><o:mi>δ</o:mi><o:msub><o:mi>u</o:mi><o:mi>z</o:mi></o:msub></o:msub></o:math>. We further describe how the transition to a state of bulk convection is influenced by the presence of the wall mode states. We use temporal and spatial scales as measures of the local state of convection and the Nusselt number <p:math xmlns:p="http://www.w3.org/1998/Math/MathML"><p:mtext>Nu</p:mtext></p:math> as representative of global transport. Our results elucidate the evolution of the wall state of rotating convection and confirm that wall modes are strongly linked with the boundary zonal flow being the robust remnant of nonlinear wall mode states. We also show how the heat transport (<q:math xmlns:q="http://www.w3.org/1998/Math/MathML"><q:mtext>Nu</q:mtext></q:math>) contributions of wall modes and bulk modes are related and discuss approaches to disentangling their relative contributions. Published by the American Physical Society 2024

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