Effect of hyperdiffusion on rotating Rayleigh-Bénard convection

We analyze the effect of a scale-dependent horizontal hyperdiffusivity in rotating Rayleigh-Bénard convection. The hyperdiffusion (HD) is parametrized by a cutoff wave number <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msub> <a:mi>k</a:mi> <a:mn>0</a:mn> </a:msub> </a:math> and a growth rate <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>q</b:mi> </b:math> to limit its effects to the smaller scales, enabling simulations to be conducted at more rapid rotation rates and buoyancy forcing (as measured by the Ekman number Ek and Rayleigh number Ra, respectively) than in conventional direct numerical simulations (DNSs). We have performed 107 simulations spanning the ranges <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mrow> <c:mi>E</c:mi> <c:mo>=</c:mo> <c:msup> <c:mn>10</c:mn> <c:mrow> <c:mo>−</c:mo> <c:mn>4</c:mn> </c:mrow> </c:msup> <c:mtext>--</c:mtext> <c:msup> <c:mn>10</c:mn> <c:mrow> <c:mo>−</c:mo> <c:mn>7</c:mn> </c:mrow> </c:msup> </c:mrow> </c:math> and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mrow> <d:mtext>Ra</d:mtext> <d:mo>=</d:mo> <d:msup> <d:mn>10</d:mn> <d:mn>6</d:mn> </d:msup> <d:mtext>--</d:mtext> <d:mn>7</d:mn> <d:mo>×</d:mo> <d:msup> <d:mn>10</d:mn> <d:mn>10</d:mn> </d:msup> </d:mrow> </d:math> and directly compared HD and DNS heat transfer (measured by the Nusselt number Nu), flow speeds (quantified by the Péclet number Pe), force balances, and boundary layer thicknesses. We identify two different effects of HD that are particularly clear when <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:msub> <e:mi>k</e:mi> <e:mn>0</e:mn> </e:msub> </e:math> is below the dominant scale of the flow <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"> <f:msub> <f:mi>k</f:mi> <f:mo>⊥</f:mo> </f:msub> </f:math> : At low supercriticality <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"> <g:mover accent="true"> <g:mi>Ra</g:mi> <g:mo>̃</g:mo> </g:mover> </g:math> , Nu, and Pe are increased compared to DNS due to a weakening of the rotational constraint; at high <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mover accent="true"> <i:mi>Ra</i:mi> <i:mo>̃</i:mo> </i:mover> </i:math> , Nu and Pe are decreased compared to DNS owing to suppression of energy at scales greater than <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msub> <k:mi>k</k:mi> <k:mo>⊥</k:mo> </k:msub> </k:math> . The thermal boundary layer thickness changes in proportion to Nu, while the mechanical boundary layer thickness is marginally affected because it is determined by a vertical balance, whereas HD is applied horizontally. Heat transfer diagnostics are more strongly affected by HD than flow diagnostics, while changing <l:math xmlns:l="http://www.w3.org/1998/Math/MathML"> <l:msub> <l:mi>k</l:mi> <l:mn>0</l:mn> </l:msub> </l:math> has a greater impact on solution behavior than changing <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> over the range of values studied. For all HD simulations with <n:math xmlns:n="http://www.w3.org/1998/Math/MathML"> <n:mrow> <n:msub> <n:mi>k</n:mi> <n:mn>0</n:mn> </n:msub> <n:mo>≥</n:mo> <n:mn>3</n:mn> <n:msub> <n:mi>k</n:mi> <n:mo>⊥</n:mo> </n:msub> </n:mrow> </n:math> , all diagnostics remain within the standard deviation of the DNS.