Dynamic Meteorology: A Review

1. Dynamic Meteorology: A Review M. D. Eastin

2. Total Vs. Partial Derivatives • Total Derivatives • The rate of change of something following a fluid element is called • the Lagrangian rate of change • Example: How temperature changes following an air parcel as is moves around • Partial Derivatives • The rate of change of something at a fixed point is called • the Eulerian rate of change • Example: The temperature change at a surface weather station • Euler’s Relation • Shows how a total derivative can be decomposed into a local rate of change • and advection terms M. D. Eastin

3. Vectors Scalar: Has only a magnitude (e.g. temperature) Vector:Has a magnitude and direction (e.g. wind) Usually represented in bold font (V) or as ( ) Unit Vectors: Represented by the letters i, j, k Magnitude is 1.0 Point in the x, y, and z (or p) directions Total Wind Vector: Defined as V = ui + vj + wk, where u, v, w are the scalar components of the zonal, meridional, and vertical wind Vector Addition/Subtraction: Simply add the scalars of each component together Vector Multiplication: Dot Product: Defined as the product of the magnitude of the vectors Results in a scalar The dot product of any unit vector with another = 0 V1+V2 = (u1+u2)i + (v1+v2)j + (w1+w2)k V1•V2= u1u2+v1v2+w1w2 i•i = j•j = k•k = 1 M. D. Eastin

4. Vectors Vector Multiplication: Cross Product: Results in a third vector that points perpendicular to the first two Follows the “Right Hand Rule” Often used in meteorology when rotation is involved (e.g. vorticity) Differential “Del” Operator: Definition: Del multiplied by a scalar (“gradient” of the scalar): Dot product of Del with Total Wind Vector (“divergence”): V1 x V2 = i(v1w2 – v2w1)+j(u2w1-u1w2)+k(u1v2-u2v1) M. D. Eastin

5. Vectors Differential “Del” Operator: Cross product of Del with Total Wind Vector (“vorticity”): Note:The third term is rotation in the horizontal plane about the vertical axis This is commonly referred to “relative vorticity” (ζ) We can arrive at this by taking the dot product with the k unit vector Dot product of Del with itself (“Laplacian” operator) If we apply the Laplacian to a scalar: M. D. Eastin

6. Vectors Euler’s Relation Revisited: If we dot multiply the gradient of a scalar (e.g. Temperature) with the total wind vector we get the advection of temperature by the wind: Recall, the total derivative of temperature can be written as (in scalar form) Or as (in vector form) upon substituting from above: M. D. Eastin

7. Equations of Motion The equations of motion describe the forces that act on an air parcel in a three-dimensional rotating system → describe the conservation of momentum Fundamental Forces: Pressure Gradient Force (PGF) → Air parcels always accelerate down the pressure gradient from regions of high to low pressure Gravitational Force (G) → Air parcels always accelerate (downward) toward the Earth’s center of mass (since the Earth’s mass is much greater than an air parcel’s mass) Frictional Force (F) → Air parcels always decelerate due to frictional drag forces both within the atmosphere and at the boundaries Apparent Forces (due to a rotating reference frame): Centrifugal Force (CE) → Air parcels always accelerate outward away from their axis of rotation Coriolis Force (CF) → Air parcels always accelerate 90° to the right of their current direction (in the Northern Hemisphere) M. D. Eastin

8. Equations of Motion The equation of motion for 3D flow can be written symbolically as: Normally, this equation is decomposed into three equations: What are each of these terms? M. D. Eastin

9. Equations of Motion The equations of motion for 3D flow: where: Total Derivative of Wind Pressure Gradient Force Gravitational Force Frictional Force Curvature Terms Coriolis Force Are all of these terms significant? Can we simplify the equations? M. D. Eastin

10. Equations of Motion • Scale Analysis: • Method by which to determine which terms in the equations can be neglected: • [Neglect terms much smaller than other terms (by several orders of magnitude)] • Use typical values for parameters in the mid-latitudes on the synoptic scale • Horizontal velocity (U) ≈ 10 m s-1 (u,v) • Vertical velocity (W) ≈ 10-2 m s-1 (w) • Horizontal Length (L) ≈ 106 m (dx,dy) • Vertical Height (H) ≈ 104 m (dz) • Angular Velocity (Ω) ≈ 10-4 s-1 (Ω) • Time Scale (T) ≈ 105 s (dt) • Frictional Acceleration (Fr) ≈ 10-3 m s-2 (Frx, Fry, Frz) • Gravitational Acceleration (G) ≈ 10 m s-2 (g) • Horizontal Pressure Gradient (∆p) ≈ 103 Pa (dp/dx, dp/dy) • Vertical Pressure Gradient (Po) ≈ 105 Pa (dp/dz) • Air Density (ρ) ≈ 1 m3 kg-1 (ρ) • Coriolis Effect (C) ≈ 1 (2sinφ, 2cosφ) • Using these values, you will find that numerous terms can be neglected….. M. D. Eastin

11. Equations of Motion The “simplified” equations of motion for synoptic-scale 3D flow: where: f = 2ΩsinΦ and Φ is the latitude This set of equations is often called the “primitive equations” for large-scale motion Note: The total derivatives have been decomposed into their local and advective terms The vertical equation of motion reduces to the hydrostatic approximation – vertical velocity can NOT be predicted using the vertical equation of motion – other approaches must be used M. D. Eastin

12. Mass Continuity Equation • The continuity equation describes the conservation of mass in a 3D system • Mass can be neither created or destroyed • Must account for mass in synoptic-scale numerical prediction • Mass Divergence Form: • Velocity Divergence Form: • Scale Analysis results in: Interpretation: Net mass change is equal to the 3-D convergence of mass into the column Form commonly used by numerical models to predict density changes with time Form commonly used by observational studies to identify regions of vertical motion M. D. Eastin

13. Mass Continuity Equation If we isolate the vertical velocity term on one side: OR Thus, changes in the vertical velocity can be induced from the horizontal convergence/divergence fields Example: Convergence near the surface (low pressure) leads to upward motion that increases with height Divergence near the surface (e.g. high pressure) leads to downward motion increasing with height L M. D. Eastin

14. Thermodynamic Equation The thermodynamic equation describes the conservation of energy in a 3D system Begin with the First Law of Thermodynamics: After some algebra…. Decomposed into local and advective components: What are each of these terms? Local change in temperature Advection of temperature Adiabatic temperature change due to expansion and contraction Diabatic temperature change from condensation, evaporation, and radiation M. D. Eastin

15. Isobaric Coordinates • Advantages of Isobaric Coordinates: • Simplifies the primitive equations • Remove density (or mass) variations that are difficult to measure • Upper air maps are plotted on isobaric surfaces • Characteristics of Isobaric Coordinates: • The atmosphere is assumed to be in hydrostatic balance • Vertical coordinate is pressure → [x,y,p,t] • Vertical velocity (ω) • ω > 0 for sinking motion • ω < 0 for rising motion • Euler’s relation in isobaric coordinates • What are the primitive equations in isobaric coordinates? M. D. Eastin

16. Isobaric Coordinates Primitive Equations (for large-scale flow) in Isobaric Coordinates: See Holton Chapter 3 for a complete description of the transformations We will be working with (starting from) these equations most of the semester!!! Zonal Momentum Meridional Momentum Hydrostatic Approximation Mass Continuity Thermodynamic Equation of State M. D. Eastin

17. Hypsometric Equation What it means: The thickness between any two pressure levels is proportional to the mean temperature within that layer Warmer layer → Greater thickness Pressure decrease slowly with height Colder layer → Less thickness Pressure decreases rapidly with height Derivation: Integrate the Hydrostatic Approximation between two pressure levels M. D. Eastin

18. Hypsometric Equation • Application: Can infer the mean vertical structure of the atmosphere: • Location/structure of pressure systems • Location/structure of jet streams • Precipitation type (rain/snow line) 1000-500-mb Thickness – 0600 UTC 22 Jan 2004 500-mb Heights – 0600 UTC 22 Jan 2004 From Lackmann (2011) M. D. Eastin

19. Geostrophic Balance • Recall the horizontal momentum equations: • Scale analysis for large-scale (synoptic) motions above the surface reveals that the total • derivatives are one order of magnitude less than the PGF and CF. • Neglect the total derivatives and do some algebra…. • The PGF exactly balances the CF • There are no accelerations acting • on the parcel (once balance is achieved) M. D. Eastin

20. Geostrophic Balance Pressure Gradient Force Geostrophic Wind Coriolis Force M. D. Eastin

21. Thermal Wind • What is Means:The vertical shear of the geostrophic wind over a layer • is directly proportional to the horizontal temperature • (or thickness) gradient through the layer • Derivation: Differentiate the geostrophic balance equations with respect to pressure • and apply the hydrostatic approximation • Characteristics: • Relates the temperature field to the wind field • Describes how much the geostrophic wind will change with height (pressure) • for a given horizontal temperature gradient • The thermal wind is the vector difference between the two geostrophic winds above • and below the pressure level where the horizontal temperature gradient resides • The thermal wind always blows parallel to the mean isotherms (or lines of constant • thickness) within a layer with cold air to the left and warm air to the right M. D. Eastin

22. Thermal Wind: Application The thermal wind can be used to diagnose the mean horizontal temperature advection within a layer of the atmosphere Warm Air Advection (WAA) (within a layer) Cold Air Advection (CAA) (within a layer) Cold Vtherm V500 V850 Warm V850 Cold V500 Vtherm Warm Geostrophic winds turn clockwise (or “veer”) with height through the layer Geostrophic winds turn counterclockwise (or “back”) with height through the layer M. D. Eastin

23. Thermal Wind: Application • International Falls, MN • Winds turn counterclockwise (“back”) • with height between 850 and 500 mb • We should expect CAAwithin the layer • Note that CAA appears to be • occurring at both 850 and 500 mb • Buffalo, NY • Winds turn clockwise (“veer”) with • height between 850 and 500 mb • We should expect WAA within the layer 500 mb 850 mb M. D. Eastin

24. Thermal Wind: Application Minneapolis / Saint Paul (MSP) We can infer WAA and CAA with a single sounding from the vertical profile of wind direction Winds are backing with height → CAA Winds are veering with height → WAA M. D. Eastin

25. Surface Pressure Tendency • What it means: The net divergence (convergence) of mass out of (in to) a column • of air will lead to a decrease (increase) in surface pressure • Derivation: Integrate the Continuity Equation (in isobaric coordinates) through the • entire depth of the atmosphere and apply boundary conditions • Characteristics: • Provide qualitative information concerning the movement (approach) of pressure systems • Difficult to apply as a forecasting technique since small errors in wind (i.e. divergence) • field can lead to large pressure tendencies • Also, divergence at one level is usually offset by convergence at another level • Note: Temperature changes in the column do not have a direct effect on the surface • pressure – they change the height of the pressure levels, not the net mass M. D. Eastin

26. Circulation and Vorticity Circulation: The tendency for a group of air parcels to rotate If an area of atmosphere is of interest, you compute the circulation Vorticity: The tendency for the wind shear at a given point to induce rotation If a point in the atmosphere is of interest, you compute the vorticity Planetary Vorticity: Vorticity associated with the Earth’s rotation Relative Vorticity: Vorticity associated with 3D shear in the wind field Only the vertical component of vorticity (the k component) is of interest for large-scale (synoptic) meteorology Absolute Vorticity: The sum of relative and planetary vorticity M. D. Eastin

27. Circulation and Vorticity Circulation: The tendency for a group of air parcels to rotate If an area of atmosphere is of interest, you compute the circulation Vorticity: The tendency for the wind shear at a given point to induce rotation If a point in the atmosphere is of interest, you compute the vorticity 500-mb Heights Absolute Vorticity (η = ζ + f) From Lackmann (2011) M. D. Eastin

28. Circulation and Vorticity Vorticity Types: Relative Vorticity (ζ) Absolute Vorticity (η = ζ + f) From Lackmann (2011) M. D. Eastin

29. Circulation and Vorticity Vorticity Types: Positive Vorticity: Associated with cyclonic (counterclockwise) circulations in the Northern Hemisphere Negative Vorticity: Associated with anticyclonic (clockwise) circulations in the Northern Hemisphere M. D. Eastin

30. Circulation and Vorticity Vorticity Types: Shear Vorticity: Associated with gradients along local straight-line wind maxima Curvature Vorticity: Associated with the turning of flow along a stream line Shear Vorticity Curvature Vorticity + _ + From Lackmann (2011) M. D. Eastin

31. Vorticity Equation Describes the factors that alter the magnitude of the absolute vorticity with time Derivation: Start with the horizontal momentum equations (in isobaric coordinates) Take of the meridional equation and subtract of the zonal equation After use of the product rule, some simplifications, and cancellations: Zonal Momentum Meridional Momentum M. D. Eastin

32. Vorticity Equation What do the terms represent? Local rate of change of relative vorticity ~10-10 Horizontal advection of relative vorticity ~10-10 Vertical advection of relative vorticity ~10-11 Meridional advection of planetary vorticity ~10-10 Divergence Term ~10-9 Tilting Terms ~10-11 What are the significant terms? → Scale analysis and neglect of “small” terms yields: M. D. Eastin

33. Vorticity Equation Physical Explanation of Significant Terms: • Horizontal Advection of Relative Vorticity • The local relative vorticity will increase (decrease) if positive (negative) relative vorticity is • advected toward the location → Positive Vorticity Advection (PVA) and • → Negative Vorticity Advection (NVA) • PVA often leads to a decrease in surface pressure (intensification of surface lows) • Meridional Advection of Planetary Vorticity • The local relative vorticity will decrease (increase) if the local flow is southerly (northerly) • due to the advection of planetary vorticity (minimum at Equator; maximum at poles) • Divergence Term • The local relative vorticity will increase (decrease) if local convergence (divergence) exists M. D. Eastin

34. Vorticity Equation Physical Explanation: Horizontal Advection of Relative Vorticity Relative Vorticity Advection Relative Vorticity (ζ) From Lackmann (2011) M. D. Eastin

35. Quasi-Geostrophic Theory • Most meteorological forecasts: • Focus on Temperature, Winds, and Precipitation (amount and type)** • Are largely a function of the evolving synoptic-scale weather patterns • Quasi-Geostrophic Theory: • Makes further simplifying assumptions about the large-scale dynamics • Diagnostic methods to estimate: Changes in large-scale surface pressure • Changes in large-scale temperature (thickness) • Regions of large-scale vertical motion • Despite the simplicity, it provides accurate estimates of large-scale changes • Will provide the basic analysis framework for remainder of the semester • Next Time…… M. D. Eastin

36. Summary • Important Dynamic Meteorology (METR 3250) Concepts: • Total / Partial Derivatives and Vector Notation • Equation of Motion (Components and Simplified Terms) • Mass Continuity Equation • Thermodynamic Equation • Isobaric Coordinates and Equations • Hypsometric Equation • Geostrophic Balance • Thermal Wind • Surface Pressure Tendency • Circulation and Vorticity • Vorticity Equation M. D. Eastin