Applications of the Basic Equations Chapter 3
Paul A. Ullrich [email protected]
Part 2: Balanced Flow Momentum Equation
One diagnostic equation for curved fow:
V 2 @ + fV = R @n
Centrifugal force Pressure gradient force
Coriolis force Question: How does this generalize the geostrophic approximation?
Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic Balance
Low Pressure
High Pressure
Flow initiated by Flow turned by pressure gradient Coriolis force
Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic Balance
Geostrophic balance: Coriolis and pressure gradient force in exact balance (equal magnitude, opposite direction).
Flow is parallel to contours: Flow follows a straight line, so the radius of curvature (R) becomes infnite.
0 V 2 @ + fV = R @n
Centrifugal force Pressure gradient force
Coriolis force
Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic Balance In Natural Coordinates
Geostrophic balance in natural coordinates @ fV = g @n
Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic Balance In Natural Coordinates
0 fVg > 0 ⇡ n North 0 + +2 Pressure n 0 Gradient Direction 0 +3 of fow
Coriolis Force South Key point: The real wind can be equal to the geostrophic wind if the geopotential contours are straight. West East
Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic Balance In Natural Coordinates
Key point: If contours are not straight then the real wind is not geostrophic.
Let’s look at this closer…
Defnition: Cyclonic fow refers to fow around a low pressure system (geopotential minimum).
Anticyclonic fow refers to fow around a high pressure system (geopotential maximum).
Paul Ullrich Applications of the Basic Equations March 2014 Question: How does curvature affect the real wind?
V 2 @ Low Pressure + fV = R @n 0 R North + n 0
n 0 +2 t 0 +3
South High Pressure
West East
Paul Ullrich Applications of the Basic Equations March 2014 Ageostrophic Wind
Equation of Motion V 2 @ (Natural Coordinates) + fV = R @n
Split Coriolis into V 2 @ geostrophic and + f(V + V )= ageostrophic R g ag @n
Use defnition of @ 2 geostrophic wind fVg = V @n = fV R ag
Key point: Centrifugal force balances with ageostrophic part of Coriolis force.
Paul Ullrich Applications of the Basic Equations March 2014 Ageostrophic Wind
Assume R > 0. Then the direction V 2 = fV of curvature is towards the low. R ag This is cyclonic motion. V 2 = fV Low R ag
Sign: > 0 > 0
R Vag < 0
But since V = Vg + Vag
V Real wind is slower than geostrophic wind around a low! Paul Ullrich Applications of the Basic Equations March 2014 Physical Perspective Pressure Pressure Gradient Gradient Vg V Coriolis Coriolis Force Force Centrifugal Force No Curvature With Curvature Vg >V Pressure gradient is the same in each case. However, with curvature less Coriolis force is needed to balance the pressure gradient. Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic & Observed Wind Upper Tropo (300mb) Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic & Observed Wind Upper Tropo (300mb) Observed: 95 knots Geostrophic: 140 knots Paul Ullrich Applications of the Basic Equations March 2014 Anticyclonic Flow Flow around a Pressure High n n t 0 0 R 0 + V 2 @ High + fV = Radius of curvature R @n points in the opposite direction of n R is negative Paul Ullrich Applications of the Basic Equations March 2014 Ageostrophic Wind Assume R < 0. Then the direction V 2 = fV of curvature is towards the high. R ag This is anti-cyclonic motion. V 2 = fV R ag Sign: < 0 < 0 Vag > 0 But since V = V + V R g ag V>Vg High Real wind is faster than geostrophic wind around a high! Paul Ullrich Applications of the Basic Equations March 2014 Physical Perspective Pressure Centripetal Gradient Force Pressure Gradient V Vg Coriolis Force Coriolis Force No Curvature With Curvature Vg Pressure gradient is the same in each case. However, with curvature more Coriolis force is needed to balance the pressure gradient + centripetal force. Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic & Observed Wind Upper Tropo (300mb) Paul Ullrich Applications of the Basic Equations March 2014 Geostrophic & Observed Wind Upper Tropo (300mb) Observed: 30 knots Geostrophic: 25 knots Paul Ullrich Applications of the Basic Equations March 2014 Natural Coordinates Summary • We found a way to describe balance between pressure gradient force, Coriolis force and curvature (centrifugal force). • We assumed friction was unimportant and only looked at fow at a particular level. • We assumed fow was on pressure surfaces. • We saw that the simplifed system can be used to describe real fows in the atmosphere. • Can we describe other fow patterns? Different scales? Different regions of the Earth? Paul Ullrich Applications of the Basic Equations March 2014 Cyclostrophic Flow Momentum equation in natural coordinates DV V 2 @ @ t + n + fVn = t + n Dt R @s @n ✓ ◆ Momentum equation in natural coordinates, component form DV @ = Dt @s Question: What about the case V 2 @ + fV = of negligible Coriolis force? R @n Paul Ullrich Applications of the Basic Equations March 2014 Cyclostrophic Flow Defnition: Cyclostrophic fow describes steady fows where centrifugal force is balanced by pressure gradient force, and Coriolis force is largely negligible. Paul Ullrich Applications of the Basic Equations March 2014 Cyclostrophic Flow Normal component of momentum equation V 2 @ + fV = R @n Cyclostrophic fow arises when V is large or R is small. In either case, the centrifugal force term is much larger than Coriolis force. Question: When might these conditions occur? Paul Ullrich Applications of the Basic Equations March 2014 Cyclostrophic Flow Cyclostrophic fow arises when V is large or R is small. In either case, the centrifugal force term is much larger than Coriolis force. Tornadoes: 100 meter radius, winds up to 50 m/s Dust Devils: 1 – 10 meter radius, winds up to 25 m/s Both of these phenomena feature small radii and (relatively) strong winds. Paul Ullrich Applications of the Basic Equations March 2014 Cyclostrophic Flow Normal component of momentum equation V 2 0 @ + fV = R @n @ Solve for V : V 2 = R @ @n V = R r @n The interior of the square root must be positive (no imaginary winds). This implies two solutions: @ @ (1) R>0, < 0 (2) R<0, > 0 @n @n Paul Ullrich Applications of the Basic Equations March 2014 Cyclostrophic Flow @ @ (1) R>0, < 0 (2) R<0, > 0 @n @n Direction Low R of fow Low R Pressure Pressure Centrifugal gradient gradient force Centrifugal force Direction of fow Counterclockwise Rotation Clockwise Rotation Paul Ullrich Applications of the Basic Equations March 2014 Cyclostrophic Flow @ @ (1) R>0, < 0 (2) R<0, > 0 @n @n Direction Low R of fow Low R Pressure Pressure Centrifugal gradient gradient force Centrifugal force Direction of fow Counterclockwise Rotation Clockwise Rotation Paul Ullrich Applications of the Basic Equations March 2014 Anticyclonic Tornado Looking up Sunnyvale, CA 4 May 1998 Paul Ullrich Applications of the Basic Equations March 2014 Question: Why can’t we have cyclostrophic fow around a high pressure system? @ @ @ (1) R>0, > 0 V = R (2) R<0, < 0 @n r @n @n Solution non-real and non-physical Direction High of fow High R R Centrifugal Centrifugal force force Solution non-real Direction and non-physical Pressure Pressure gradient of fow gradient Paul Ullrich Applications of the Basic Equations March 2014 Inertial Flow Defnition: Inertial fow describes steady fows where centrifugal force is balanced by Coriolis force and pressure gradients are largely negligible. 0 V 2 @ Uninteres ng + fV = Solutions V =0 R @n and V = fR Since V > 0, must have R < 0 Paul Ullrich Applications of the Basic Equations March 2014 Inertial Flow V = fR Inertial motion is always anti-cyclonic (clockwise in northern hemisphere). Since the circumference of the circle is C =2⇡R R The period of rotation is C 2⇡R ⇡ P = = = Coriolis Centrifugal V fR ⌦ sin Force force | | But since Ω is rotation rate of the Earth Direction 1 day of fow P = 2 sin | | Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow Momentum equation in natural So far we have investigated: coordinates, component form • Balance between pressure gradient DV @ force and Coriolis (Geostrophic = Balance). Dt @s • V 2 @ Balance between pressure gradient + fV = force and Centrifugal force R @n (Cyclostrophic fow). • Balance between centrifugal force and Coriolis force (Inertial fow). • We now consider balance between all three terms in the normal momentum equation (Gradient fow). Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow V 2 @ + fV = R @n Observe this is simply a quadratic equation in V @ V 2 + fRV + R =0 @n Quadra c Equa on 1 @ V = fR ( fR)2 4R 2 " ± r @n # fR (fR)2 @ Gradient fow velocity V = R 2 ± r 4 @n Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow The solution V must be real and non-negative. fR (fR)2 @ fR (fR)2 @ + R R 2 r 4 @n 2 r 4 @n @ @ > 0,R>0 V<0 > 0,R>0 V<0 @n @n @ @ > 0,R<0 Anomalous Low > 0,R<0 V<0 @n @n @ @ < 0,R>0 Normal Low < 0,R>0 V<0 @n @n @ @ < 0,R<0 Anomalous High < 0,R<0 Normal High @n @n Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow Solutions for Lows @ @ (1) < 0,R>0 (2) > 0,R<0 @n @n Normal Low Anomalous Low Direction Pressure Low Low R of fow gradient R Pressure Coriolis Coriolis Centrifugal gradient force force force Centrifugal force Direction of fow Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow Solutions for Highs @ @ (1) < 0,R<0 (2) < 0,R<0 @n @n Normal High Anomalous High High R High R Pressure Pressure Coriolis Coriolis gradient gradient force force Centrifugal Centrifugal Direction force Direction force of fow of fow Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow Implications fR (fR)2 @ V = R 2 ± r 4 @n The discriminant (the term inside the square root) must be positive for a stable solution to exist. (fR)2 @ R 0 4 @n R>0 R<0 This is a constraint on the pressure @ f 2R @ f 2R gradient that must hold for any @n 4 @n 4 steady fow. Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow Implications @ Consider < 0 @n Normal Low Normal / Anomalous High Direction Low R of fow High n R t R<0 Normal direction n R>0 t Normal Direction direction of fow Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow Normal Low Normal / Anomalous High @ @ < 0,R>0 < 0,R<0 @n @n @ f 2R @ f 2R @n 4 @n 4 @ f 2 R Always satisfed! Trouble! Constraint implies | | @n 4 In anti-cyclonic gradient fow, pressure must go to zero faster than R goes to zero. Hence high pressure regions must have relatively fat pressure gradient. Paul Ullrich Applications of the Basic Equations March 2014 Gradient Flow Normal / Anomalous Flows • Normal fows are observed all the time • Normal highs (atmospheric blocks) tend to have slower winds than normal lows (tropical cyclones, extratropical cyclones) since the pressure gradient is bounded. • Normal lows are storms; normal highs are fair weather. • Anomalous fows are not often observed • Anomalous highs have been reported in the tropics. • Anomalous lows are “strange” – quoting Holton, “clearly not a useful approximation.” However, these do appear for very small values of R (ie. Tornados). Paul Ullrich Applications of the Basic Equations March 2014