Investigating Ocean DeepConvection Using Multi-ScaleAsymptoticsIan GroomsDepartment of Applied MathematicsUniversity of ColoradoWork Performed With and Under the Direction ofKeith Julien
Outline●Review– Ocean Deep Convection (ODC): Observations– Derivation of UNH-QGE●●●Adding Multiple ScalesSeveral Equation SetsFuture Work
ODC: Observations●●●●Horizontal Plume Scale < 1kmEddy Scale 10 kmGyre Scale 100 kmPlume Depth 2-5 km
Upright Non-Hydrostatic Quasi-Geostrophic EquationsStart with the Rotating Boussinesq Equations.Nondimensionalize using different horizontal andvertical scales, letting the aspect ratio (tall) be asmall parameter that scales like the Rossbynumber. (As in Keith Julien's talk)A z =L/ H ~
The UNH-QGE∂ t wJ [ , w]D = ∇ ⊥ 2w∂ t ∇ ⊥ 2 J [ , ∇ ⊥ 2 ]−D w=∇ ⊥ 4 ∂ t J [ ,]w D = −1 ∇ ⊥ 2 ∂ TD w = −1 D 2
Adding Multiple ScalesThe UNH-QGE describe the plume scale, butaren't well suited to examining the dynamics of theocean gyres where deep convection takes place.In order to capture the evolution of the gyre andthe baroclinic eddies that form along its edge, weadd another horizontal scale.
Adding Multiple Scales∇ ∇ ⊥A ⊥ ∇ ⊥A z z D∂ t ∂ tA T∂ Tf = limt , V ∞1t V ∫ f dt dVf X ,Y , Z ,T , x , y ,t=f X ,Y , Z ,T f ' X ,Y , Z ,T , x , y ,tf '=0f g= f g f ' g '
Adding Multiple Scales: The MeanEquationsA T ∂ T uA ⊥ ∇ ⊥ ⋅u uA z Dw uRo −1 z×u=−P A ⊥ ∇ ⊥ A z z D p zRe −1 A ⊥ ∇ ⊥ z A z D 2 uA T ∂ TA ⊥ ∇ ⊥ ⋅uA z Dw =Pe −1 A ⊥ ∇ ⊥ z A z D 2 A ⊥ ∇ ⊥ ⋅uA z D w=0
Adding Multiple Scales: TheDistinguished LimitThe choice of relationships between the varioussmall parameters is called a distinguished limit.We make this choice consistently withobservations of the phenomena we wish todescribe, but also for the purpose of arriving at aclosed system of equations. In order for the meanequations to be in geostrophic and hydrostaticbalance, and to have feedbacks to and from thefluctuating equations, we choose as follows.Ro≡ , A T ~A ⊥ ~ 2 , A z ~ , P~ −3 , ~ −2
Adding Multiple Scales: AsymptoticSeriesTo arrive at a closed system for the mean andfluctuating velocities and potential temperature, itis not sufficient to simply examine the leadingorder balances (as we have seen in the derivationof both traditional QG and of the UNH-QGE). Wemust also look at corrections to leading orderbalances by using asymptotic series.u=u 0 u 1 O 2 = 0 1 O 2 u'=u 0 'u 1 ' O 2 '= 0 ' 1 'O 2
Adding Multiple Scales: The FinalResultUsing the idea of an asymptotic series, balancingthe equations order by order, and imposingsolvability conditions now allow us to deriveequations for the mean and fluctuating flows (notshown). The result links the PlanetaryGeostrophic Equations describing the mean flowto the UNH-QGE describing the fluctuations.
Adding Multiple Scales: The FinalResultFluctuating equations:∂ t w ' u⋅∇ ⊥ w ' J [' , w ' ]D' = −1 '∇ ⊥ 2∂ t ∇ ⊥ 2 'u⋅∇ ⊥ ∇ ⊥ 2 ' J [' , ∇ ⊥ 2 ' ]−D w '=∇ ⊥ 4 '∂ t 'u⋅∇ ⊥ 'J [' , ' ]w ' D = −1 ∇ ⊥ 2 'Mean Equations:∂ Tu⋅∇ ⊥Dw ' ' = −1 D 2 z×u=− ∇ ⊥ pD p= D w=0w '
Several Equation Sets: PGE + QGEFluctuating Equations:∂ t q' u⋅∇ ⊥ q'J [' , q' ]=0q'=∇ 2 '⊥ '− f Y DD p'= f ' , u'=−∇×' z , '= f D 'Mean Equations∂ Tu⋅ ∇ ⊥ w D ∇ ⊥ ⋅u' ' =D D −1 ' u ' ⋅∇ ⊥D =D pf ×u=− ∇ ⊥ p∇ ⊥ ⋅uD w=0
Several Equation Sets: QGB + QGEFluctuating Equations:The UNH-QGEMean Equations:∂ T u⋅∇ ⊥ =−z⋅[1∇ ⊥ × ∇ ∫ ⊥0∂ Tu⋅ ∇ ⊥Dw ' ' =D 2 =z⋅[ ∇ ⊥ ×u ]=− ∇ ⊥2D p=0pu' u' dz]
Future Work●●●Do some computationsAdd the effects of wind and topographyDo something about the equator (PGE breakdown at the equator)