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The Quasi-Geostrophic Omega Equation (without friction and diabatic terms) We will now develop the Trenberth (1978)* modification to the QG Omega equation THE TRENBERTH (1978) INTERPRETATION *Trenberth, K.E., 1978: On the Interpretation of the Diagnostic Quasi-Geostrophic Omega Equation. Mon. Wea. Rev., 106, 131–137 Trenberth (1978) argued that carrying out all of the derivatives on the RHS on the Equation could simplify the forcing function for . PROBLEM: TERM 1 and 2 on the RHS are often large and opposite leading to ambiguity about the sign and magnitude of when analyzing weather maps

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QG OMEGA EQUATION: EXPAND THE ADVECTION TERMS: USE THE EXPRESSIONS FOR THE GEOSTROPHIC WIND AND GEOSTROPHIC VORTICITY: To Get:

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EXPAND ALL THE DERIVATIVES THEN USE THE JACOBIAN OPERATOR TO SIMPLIFY NOTATION RESULT:

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Same term: opposite sign Deformation Terms in Sutcliff eqn Trenberth: Ignore deformation terms (removes frontogenetic effects) Opposite term: opposite sign = same term Approximate last term = 2 last term

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Expand Jacobian terms: This result says that large scale vertical motions can be diagnosed by Examining the advection of absolute vorticity by the thermal wind RECALL SUTCLIFF’S EQUATION SAME INTERPRETATION!!

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The Geostrophic paradox Confluent geostrophic flow will tighten temperature gradient, leading to an increase in shear via the thermal wind relationship…….. ……but advection of geostrophic momentum by geostrophic wind decreases the vertical shear in the column so…geostrophic flow destroys geostrophic balance!

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y momentum equation (QG) Thermodynamic energy equation (QG) For the moment, let’s ignore the ageostrophy (no u ag and no ) The geostrophic paradox: a mathematical interpretation Take vertical derivative of first equation Let’s look at this equation

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Expand the derivative: Substitute using the thermal wind relationship: to get: Remember equation in blue box

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y momentum equation (QG) Thermodynamic energy equation (QG) For the moment, let’s ignore the ageostrophy (no u ag and no ) The geostrophic paradox: a mathematical interpretation Take x derivative of second equation Now let’s look at this equation

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Expand the derivative and use vector notation: Now recall first saved equation: Let’s take these two blue boxed equations and compare them…..

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Following the geostrophic wind the magnitude of the temperature gradient and the vertical shear have opposite Tendencies TIGHTENING THE TEMPERATURE GRADIENT WILL REDUCE THE SHEAR! thermal wind balance

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The Geostrophic paradox: RESOLUTION A separate “ageostrophic circulation” must exist that restores geostrophic balance that simultaneously: 1)Decreases the magnitude of the horizontal temperature gradient 2) Increases the vertical shear

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The Q-Vector interpretation of the Q-G Omega Equation (Hoskins et al. 1978) From consideration of the geostrophic wind, we derived these equations: Let’s denote the term on the RHS: y momentum equation (QG) Thermodynamic energy equation (QG) If we start with our original equations, below, and perform the same operations as before, but with the ageostrophic terms included….

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We arrive at: With ageostrophic terms Only geostrophic terms Note that the additional terms represent the ageostrophic circulation that works to reestablish geostrophic balance as air accelerates in unbalanced flow.

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With ageostrophic terms Let’s multiply the bottom equation by -1 and add it to the top equation, recalling that Let’s do the same operations with the x equation of motion and the thermodynamic equation. If we do, we find that:

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Let’s do the same operations with the x equation of motion and the thermodynamic equation. Where: and:

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A B Take Substitute continuity equation And use vector notation to get:

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COMPARE THIS EQUATION WITH THE TRADITIONAL QG EQUATION! We can write the Q-vector form of the QG equation as: Where the components of the Q vector are

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Using the hydrostatic relationship, we can write Q more simply as: or in scalar notation as

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First note that if the Q vector is convergent Therefore air is rising when the Q vector is convergent

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Let’s go back to our jet entrance region Note that there is no in this particular jet Q convergence Q divergence The Q vectors capture the sense of the ageostrophic circulation and allow us to see where the rising motion is occurring

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Q convergence Q divergence The Q vectors capture the sense of the ageostrophic circulation and allow us to see where the rising motion is occurring Q vectors diagnose a thermally direct circulation Adiabatic cooling of rising warm air Adiabatic warming of sinking cold air Counteracts the tendency of the geostrophic temperature advection in confluent flow Under influence of Coriolis force, horizontal branches tend to increase shear Counteracts the tendency of the geostrophic Momentum advection in the confluent flow Resolution of the Geostrophic Paradox

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A natural coordinate version of the Q vector (Sanders and Hoskins 1990) Consider a zonally oriented confluent entrance region of a jet where Use non-divergence of geostrophic wind or

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A natural coordinate version of the Q vector (Sanders and Hoskins 1990) Consider a meridionally oriented confluent entrance region of a jet where Use non-divergence of geostrophic wind or

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A natural coordinate version of the Q vector (Sanders and Hoskins 1990) Using these two expressions, let’s adopt A natural coordinate expression for Q Adopt a coordinate system where is directed along the isotherms is directed normal to the isotherms Q vector oriented perpendicular to the vector change in the geostrophic wind along the isotherms. Magnitude proportional to temperature gradient and inversely proportional to pressure.

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Simple Application #1 Train of cyclones and anticyclones At center of highs and lows: Black arrows: Gray arrows = Bold arrows = sinking motion rising motion Note also that because of divergence/convergence, train of cyclones and anticyclones propagates east along direction of thermal wind

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Simple Application #2 Pure deformation flow with a temperature gradient Along axis of dilitation Black arrows: Gray arrows = Bold arrows = sinking motion rising motion Increases toward east

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Simple Application #3 Homogeneous warm advection No variation in Black arrows: Gray arrows = Bold arrows = along an isotherm No heterogeneity in the warm advection field = No rising motion!

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Note that the Q vector form of the QG -equation contains the deformation terms (unlike the Sutcliff and Trenberth forms) And combines the vorticity and thermal advection terms into a single diagnostic (unlike the traditional QG -equation) Sutcliff/Trenberth approximation Deformation term contribution to

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The along and across-isentrope components of the Q vector Begin with the hydrostatic equation in potential temperature form where: And the definition of the Q vector: (which is constant on an isobaric surface) Substituting: This expression is equivalent to:

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The Q-vector describes the rate of change of the potential temperature along The direction of the geostrophic flow Let’s consider separately the components of Q along and across the isentropes

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Is parallel to and can only affect changes in the magnitude of Is perpendicular to and can only affect changes in the direction of

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Returning to QG equation Components of vertical motion can be distributed in couplets across (transverse to) the thermal wind (mean isotherms) and along (shearwise) the thermal wind. We will see later that the transverse component of Q is related to the dynamics of frontal zones.

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The quasi-geostrophic omega equation (see Bluestein, Volume I; eq. 5.6.14 ( 2 + (f 0 2 / ) 2 /∂p 2 ) = (-f 0 / ) / p{-v g p ( g + f)}

The quasi-geostrophic omega equation (see Bluestein, Volume I; eq. 5.6.14 ( 2 + (f 0 2 / ) 2 /∂p 2 ) = (-f 0 / ) / p{-v g p ( g + f)}

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