Scale interactions between the meso‐ and synoptic scales and the impact of diabatic heating

Abstract

<title xmlns:mml="http://www.w3.org/1998/Math/MathML">Abstract</title><p xmlns:mml="http://www.w3.org/1998/Math/MathML" xml:lang="en">For both the meso‐ and synoptic scales, reduced mathematical models give insight into their dynamical behaviour. For the mesoscale, the weak temperature gradient approximation is one of several approaches, while for the synoptic scale the quasigeostrophic theory is well established. However, the way these two scales interact with each other is usually not included in such reduced models, thereby limiting our current perception of flow‐dependent predictability and upscale error growth. Here, we address the scale interactions explicitly by developing a two‐scale asymptotic model for the meso‐ and synoptic scales with two coupled sets of equations for the meso‐ and synoptic scales respectively. The mesoscale equations follow a weak temperature gradient balance and the synoptic‐scale equations align with quasigeostrophic theory. Importantly, the equation sets are coupled via scale‐interaction terms: eddy correlations of mesoscale variables impact the synoptic potential vorticity tendency and synoptic variables force the mesoscale vorticity (for instance due to tilting of synoptic‐scale wind shear). Furthermore, different diabatic heating rates—representing the effect of precipitation—define different flow characteristics. With weak mesoscale heating relatable to precipitation rates of <mml:math id="jats-math-1" display="inline" overflow="scroll"><mml:mrow><mml:mi>𝒪</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>6</mml:mn><mml:mspace width="0.3em"/><mml:mtext>mm</mml:mtext><mml:mo>·</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">h</mml:mi></mml:mrow><mml:mrow><mml:mo form="prefix">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>, the mesoscale dynamics resembles two‐dimensional incompressible vorticity dynamics and the upscale impact of the mesoscale on the synoptic scale is only of a dynamical nature. With a strong mesosocale heating relatable to precipitation rates of <mml:math id="jats-math-2" display="inline" overflow="scroll"><mml:mrow><mml:mi>𝒪</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>60</mml:mn><mml:mspace width="0.3em"/><mml:mtext>mm</mml:mtext><mml:mo>·</mml:mo><mml:msup><mml:mrow><mml:mi mathvariant="normal">h</mml:mi></mml:mrow><mml:mrow><mml:mo form="prefix">−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math>, divergent motions and three‐dimensional effects become relevant for the mesoscale dynamics and the upscale impact also includes thermodynamical effects.</p>

<p xmlns:mml="http://www.w3.org/1998/Math/MathML" xml:lang="en">We develop a two‐scale asymptotic model for the meso‐ and synoptic scales following a weak temperature gradient balance and quasigeostrophic theory, but with explicit scale interactions and dependent on the mesoscale diabatic heating. With weak mesoscale heating, the mesoscale dynamics resembles 2D incompressible vorticity dynamics and the upscale impact on the synoptic scale is only of a dynamical nature. With strong mesoscale heating, divergent motions and 3D effects become relevant for the mesoscale and the upscale impact also includes thermodynamical effects. <boxed-text position="anchor" id="qj4456-blkfxd-0001" content-type="graphic" xml:lang="en"><graphic position="anchor" id="jats-graphic-1" xlink:href="urn:x-wiley:00359009:media:qj4456:qj4456-toc-0001">