Vertical Shear, Diapycnal Shear and the Gradient Richardson Number
In Cartesian coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,<...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-10-01
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| Series: | Oceans |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2673-1924/5/4/45 |
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| Summary: | In Cartesian coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> the gradient Richardson number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>i</mi></mrow></semantics></math></inline-formula> is the ratio between the square of the buoyancy frequency <i>N</i> and the square of the vertical shear <i>S</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>i</mi><mo>=</mo><mrow><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>/</mo><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mo>−</mo><mfenced separators="|"><mrow><mrow><mrow><mi>g</mi></mrow><mo>/</mo><mrow><mi>ρ</mi></mrow></mrow></mrow></mfenced><mtext> </mtext><mrow><mrow><mo>∂</mo><mi>ρ</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><mo>∂</mo><mi>u</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mrow></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><mo>∂</mo><mi>v</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mrow></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula>, with <i>ρ</i> potential density, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="|"><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow></mfenced></mrow></semantics></math></inline-formula> the horizontal velocity components and <i>g</i> gravity acceleration. In isopycnic coordinates <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>ρ</mi></mrow></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>i</mi></mrow></semantics></math></inline-formula> is expressed as the ratio between <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><msup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≡</mo><mi>N</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> and the squared diapycnal shear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><mi>ρ</mi></mrow><mo>/</mo><mrow><mi>g</mi></mrow></mrow></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mfenced open="[" close="]" separators="|"><mrow><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><mo>∂</mo><mi>u</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>ρ</mi></mrow></mrow></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mrow><mrow><mo>∂</mo><mi>v</mi></mrow><mo>/</mo><mrow><mo>∂</mo><mi>ρ</mi></mrow></mrow></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfenced></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>i</mi><mo>=</mo><mrow><mrow><msup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>/</mo><mrow><msup><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow></semantics></math></inline-formula>. This could suggest that a decrease (increase) in stratification brings a decrease (increase) in dynamic stability in Cartesian coordinates, but a stability increase (decrease) in isopycnic coordinates. The apparently different role of stratification arises because <i>S</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow></semantics></math></inline-formula> are related through the stratification itself, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>=</mo><mrow><mrow><mi>S</mi></mrow><mo>/</mo><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mrow></mrow></semantics></math></inline-formula>. In terms of characteristic times, this is equivalent to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>≡</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mo>=</mo><mrow><mrow><msup><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>o</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow><mo>/</mo><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></mrow></mrow></semantics></math></inline-formula>, which is interpreted as a critical dynamic time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi></mrow></semantics></math></inline-formula> that equals the buoyancy period <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>o</mi></mrow></msub><mo>≡</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula> normalized by the ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow><mo>/</mo><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>o</mi></mrow></msub></mrow></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>t</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>=</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></semantics></math></inline-formula> is the deformation time. Here we follow simple arguments and use field data from three different regions (island shelf break, Gulf Stream and Mediterranean outflow) to endorse the usefulness of the isopycnal approach. In particular, we define the reduced squared diapycnal shear <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>M</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> and compare it with the reduced squared vertical <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula>, both being positive (negative) for unstable (stable) conditions. While both <i>Ri</i> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> remain highly variable for all stratification conditions, the mean <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> values approach <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> with increasing stratification. Further, the field data follow the relation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mrow><mrow><mfenced separators="|"><mrow><mn>1</mn><mo>−</mo><mi>R</mi><mi>i</mi></mrow></mfenced></mrow><mo>/</mo><mrow><mfenced separators="|"><mrow><msup><mrow><mi>N</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>R</mi><mi>i</mi></mrow></mfenced></mrow></mrow></mrow></semantics></math></inline-formula>, with a subcritical <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mi>i</mi><mo>=</mo><mn>0.22</mn></mrow></semantics></math></inline-formula> for both the island shelf break and the Mediterranean outflow. We propose <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>ρ</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></mrow></semantics></math></inline-formula> to be good indexes for the occurrence of effective mixing under highly stratified conditions. |
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| ISSN: | 2673-1924 |