GEF 1100 Klimasystemet. Chapter 6: The equations of fluid motion

GEF1100 Autumn 2014 16.09.2014 GEF 1100 Klimasstemet Chapter 6: The equations of fluid motion Prof. Dr. Kirstin Krüger (MetOs, UiO since October 2013) 1

Who am I? observations Prof. Dr. Kirstin Krüger Research interests: kkrueger@geo.uio.no Stratospheric oone Dnamics of the stratosphere Influence of volcanic eruptions on climate/ earth sstem Stratosphere-ocean interactions Volcanic ecursion climate modelling ship epeditions

Lecture Outline Ch. 6 Ch. 6 - The equations of fluid motions 1. Motivation 2. Basics 2.1 Lagrange - Euler 2.2* Mathematical add on 3. Equation of motion for a nonrotating fluid 3.1 Forces 3.2 Equations of motion 3.3 Hdrostatic balance 4. Conversation of mass 5. Thermodnamic equation 6. Equation of motion for a rotating fluid 6.1 Forces 6.2 Equations of motion 7. Take home messages *Addition, not in book. 3

1. Motivation Motivation Marshall and Plumb (2008) - Derivation of the general equation of motion - Different horiontal wind forms - General circulation of atmosphere/ ocean 4

2. Basics Equations of fluid motions Hdrodnamics* focuses on moving liquids and gases. Hdrodnamics and thermodnamics are the foundations of meteorolog and oceanograph. Equations of motion for the atmosphere and ocean are constituted b macroscopic conservation laws for substances, impulses and internal energ (hdrodnamics and thermodnamics). The characteristics of ver small volume elements are looked at (e.g. overall mass, average speed). Phenomenological derivation of hdrodnamics based on the conservation of momentum, mass and energ! *hdro (greek): water 5

2. Basics Differentiation following the motion - Lagrange and Euler perspectives Lagrange perspective: The control volume is an air or seawater parcel that alwas consists of the same particles and which moves along with the wind or the current. Euler perspective: The control volume is anchored stationar within the coordinate sstem, e.g. a cube through which the air or the seawater flows through. Lagrange T(X t ) χ=x t T(X t+1 ) Parcel of air/ trajector Euler Fied bo in space T( 1,t 1 ), u( 1,t 1 ) T( 2,t 2 ), u( 2,t 2 ) 6

2. Basics Differentiation following the motion - Lagrange Euler derivatives Eample: wind blows over hill, cloud forms at the ridge of the lee wave; stead state assumption eq. cloud does not change in time. Lagrangian derivative (after Lagrange; 1736-1813): CC tt fied particle 0, C = C(,,,t): quantit, i.e., cloud amount : partial derivatives; other variables tt are kept fied during the differentiation Eulerian derivative (after Euler; 1707-1783): fied point in space = 0, Joseph-Louis Lagrange (I) 7 Leonhard Euler (CH)

2. Basics Differentiation following the motion mathematicall For arbitrar small variations: δδcc = CC tt δδtt + CC δδcc ffffffffff pppppppppppppppp = δδ + CC CC tt + uu CC δδ + CC δδ + vv CC + w CC δ: Greek small delta, infinitesimal small sie Insert δδ = uδδtt, δδ = vδδtt, δδ = wδδtt δδtt In limit of small variations: CC tt ffffffffdd pppppppppppppppp = CC tt + uu CC + vv CC + w CC = DDCC DDtt DD DDtt : DD DDtt = tt + uu + vv + w tt + uu 8

2. Basics Differentiation following the motion mathematicall Lagrangian or total derivative (after Lagrange; 1736-1813): DD DDtt fied particle tt + uu tt + vv + w + uu (6-1) uu = (u,v,w) velocit vector,, gradient operator, called nabla Time rate of change of some characteristic of a particular element of fluid, which in general is changing its position. Eulerian derivative (after Euler; 1707-1783): tt fied point Time rate of change of some characteristic at a fied point in space but with constantl changing fluid element because the fluid is moving. 9

2. Basics Nomenclature summar δ : Greek small delta, infinitesimal small sie, e.g. δv:= small air volume. : partial derivative of a quantit with respect to d d a coordinate, e.g. : total differential, derivative of a quantit with respect to all dependent coordinates : Lagrangian (or total) derivative 10

2. Basics Differentiation following the motion - Lagrangian perspective Trajectories started @850 hpa: - top: between 26.04.-29.04.1986 Eample: nuclear reactor accident Chernobl on 26.04.1986, 01:30 am Trajectories (=flight path) are used for the prediction of air movement. - bottom: between 30.04.-05.05.1986 Kraus (2004) 11

2. Basics Lagrange and Euler differentiation Think about eamples in our everda life for Euler s and Lagrange s differentiation. Give three eamples. 12

2.2 Basics - Mathematical add on September 16, 2014 2.2 Basics - Mathematical add on - Vector operations - Nabla operator Modified slides from Clemens Simmer, Universit Bonn 13

2.2 Mathematical add on Vector operations - Multiplication with a scalar a - av = va = av i = v a i = v av v av = av av With a scalar multiplication the vector stas a vector. Each element of a vector is multiplied individuall with a scalar a. The vector etends (or shortens) itself b the factor a. Convention: With the multiplication scalar vector we don t use a point (like with scalar scalar). 14

Vector operations - Scalar product - i i i i i i i v f v f v f f v f v f v f f f v v v v f f v = = + + = = = The scalar product of two vectors is a scalar. It is multiplied component-wise, then added. Convention: The scalar product is marked b a multiplication sign ( ). It is at a maimum with parallel vectors and disappears when the vectors are perpendicular to each other. f α v cosα f v f v = 2.2 Mathematical add on 15

Vector operations vector product b c a c b a b a c b a b a b a b a b a b a b a b a k b a b a j b a b a i b b b a a a k j i ba b a c = = = + = = = =, sin ) ( ) ( ) ( α a b The vector product of two vectors is again a vector. Convention: The vector product or cross product is marked b an. If is turned to on the fastest route possible, the vector (from the vector product) points in the direction in which a right-hand screw would move (right hand rule). With parallel vectors, it disappears, and it reaches its maimum when the two vectors are at right angles to each other! 16 2.2 Mathematical add on

2.2 Mathematical add on with Nabla operator spatial gradient Nabla - Operator : = = i Nabla is a (vector) operator that works to the right. It onl results in a value if it stands left to an arithmetic epression. If it stands to the right of an arithmetic epression, it keeps its operator function (and waits for application). Graphicall: - Pooling of the spatial gradients towards the spatial coordinate aes vector - Gradient points towards the largest increase of the quantit. - Amount is the magnitude of the derivative pointing towards the largest increase. 17

T T grad T T T T = = w v u w v u u div u + + = = = u v w u v w w v u k j i w v u u rot u Vector product, vector Scalar product, scalar Product with a Scalar, vector Nabla operator 2.2 Mathematical add on 18

Nabla operator algorithms: = = = T T T T T T T T T w v u w v u u w v u w v u div u u + + = = + + = 2.2 Mathematical add on Note: Nabla is a (vector) operator, i.e. the sequence must not be changed here! 19

2. Basics State variables for atmosphere and ocean u Wind = (u,v,w) in m/s T Temperature in C or K p Pressure in N/m 2 = 1 Pa (1 hpa = 10 2 Pa = 1 mbar) Specificall for the atmosphere: q Specific humidit in kg water vapour/kg moist air Specificall for the ocean: S Salinit in kg salt/kg seawater (often the salt content is epressed in practical salinit unit: psu = kg salt/1,000 kg seawater). 20

3. Equations of motion - nonrotating fluid 3. Equations of motion - nonrotating fluid I. Laws of motion for a fluid parcel in -,-,directions (appling Newtons 1. and 2. law) II. Conservation of mass III. Law of thermodnamics (including motion) 5 equations for the evolution of the fluid (5 unknowns) 21

3. Equations of motion - nonrotating fluid Momentum conservation The conservation of momentum is represented b Newton s first law ( Le prima ): (momentum = mass velocit) momentum = const. at absence of forces Momentum sentence ( Le secunda ): temporal change in momentum = Force (mass acceleration = Force) M du = F dt t: time; M: mass; u: velocit vector F: Force vector 22

3. Equations of motion - nonrotating fluid Forces on a fluid parcel ρρ δδδδ δδδδ δδδδ DDuu DDDD = F (Eq. 6-2) DDuu = uu + u uu DDDD = uu + (uu ) uu uu uu +v +w ρρ: densit (δδδδ, δδδδ, δδδδ): fluid parcel with infinitesimal dimensions δδmm: mass of the parcel; δδmm = ρρ δδδδ δδδδ δδδδ uu: parcel velocit F: net force Marshall and Plumb (2008) 23

3. Equations of motion - nonrotating fluid Gravitational force F gravit = g δδmm F gravit = g ρρ δδδδ δδδδ δδδδ (Eq. 6-3) F gravit : Gravitational force M: mass g: gravit acceleration, g = const. ρρ: densit : unit vector in upward direction Marshall and Plumb (2008) 24

3. Equations of motion - nonrotating fluid Pressure gradient force F(A) = p AA = pp ( δδδδ,, ) δδδδ δδδδ 2 F(B) = p BB = pp ( + δδδδ,, ) δδδδ δδδδ F = F(A) + F(B) Appl Talor epansion (A.2.1) at midpoint of parcel and neglect small terms (see book): 2 F pressure = FF, FF, FF = pp, pp, pp δδδδ δδδδ δδδδ Marshall and Plumb (2008) = pp δδδδ δδδδ δδδδ (Eq. 6-4) pp: pressure 25

3. Equations of motion - nonrotating fluid Friction force - Friction force operates on the earth s surface, - the greater the surface roughness, the higher the friction force, - the greater the wind velocit, the higher the friction force, - friction force takes effect in vertical distances of ~ 100 m up to 1,000 m ( atmospheric boundar laer Chapter 7). F friction = ρρ F δδδδ δδδδ δδδδ (Eq. 6-5) F: frictional force per unit mass ρρ: densit 26

3. Equations of motion - nonrotating fluid Equations of motion All three forces together in Eq. 6-2 gives: ρρ δδ δδ δδ DDuu DDDD = F pppppppppppppppp + F gggggggggggggg + F ffffffffffffffff Re-arranging leads to equation of motion for fluid parcels: DDuu DDDD + 1 ρρ pp + g = F (Equation 6-6) ρρ: densit u: velocit vector F: Force vector F: frictional force per unit mass p: pressure g: gravit acceleration : unit vector in -direction 27

3. Equations of motion - nonrotating fluid Equations of motion component form (Eq. 6-6) in Cartesian coordinates: uu tt vv ww + uu + uu vv + uu ww + vv + w + 1 pp ρρ vv + vv + w vv + 1 ρρ ww + vv + w ww + 1 + g = F ρρ = F (6-7a) = F (6 7b) (6-7c) uu: onal velocit vv:meridional velocit w : vertical velocit ρρ: densit F: frictional force per unit mass p: pressure g: gravit acceleration 28

3. Equations of motion - nonrotating fluid Hdrostatic balance If friction F and vertical acceleration DDww DDDD are negligible, we derive from the vertical eq. of motion (6-7c) the hdrostatic balance (Ch.3, Eq.3-3): = - ρρg (Eq. 6-8) Balance between vertical pressure gradient and gravitational force! Note: This approimation holds for large-scale atmospheric and oceanic circulation with weak vertical motions. 29

GEF1100 Autumn 2014 18.09.2014 GEF 1100 Klimasstemet Chapter 6: The equations of fluid motion Prof. Dr. Kirstin Krüger (MetOs, UiO) 30

4. Conservation of mass Conservation of mass continuit equation Conserved quantit: A quantit of which the derivative equals ero. Conservation of mass for liquids is also called continuit equation. 32

4. Conservation of mass Conservation of mass Marshall and Plumb (2008) 33

4. Conservation of mass Conservation of mass -1 Mass continuit requires: tt (ρρ δδδδ δδ δδ) = δδ δδ δδ = (net mass flu into the volume) mass flu in -direction per unit time into volume ρρρρ 1 2 δδδδ,, δδ δδ mass flu in -direction per unit time out of volume ρρρρ + 1 2 δδδδ,, δδ δδ Marshall and Plumb (2008) 34

4. Conservation of mass Conservation of mass -2 net mass flu in -direction into the volume is then (emploing Talor epansion A.2.2): ρρρρ δδ δδ δδ, net mass flu in - and - direction accordingl (see book). Net mass flu into the volume: ρρuu δδ δδ δδ, substituting into mass continuit: Marshall and Plumb (2008) (ρρ δδδδ δδ δδ) = ρρuu δδ δδ δδ 35

4. Conservation of mass Conservation of mass -3 leads to equation of continuit: tt + ρρuu = 0 (Eq. 6-9), which has the form of conservation law: CCCCCCCCCCCCCCCCCCCCCCCCCC tt + ffffffff = 0. Marshall and Plumb (2008) Using D/Dt (Eq. 6-1) and ρρuu =ρρρρ uu + uu (see A.2.2) we can rewrite: DDρρ DDtt + ρρρρ uu = 0 (Eq. 6-10) 36

4. Conservation of mass Approimation for continuit equation: incompressible - compressible flows Incompressible flow (e.g. ocean): uu = uu + vv + ww = 0 (Eq.6-11) Compressible flow (e.g. air) epressed in pressure coordinate p : pp uu pp = uu + vv + ww pp = 0 (Eq.6-12) 37

5. Thermodnamic equation Thermodnamic equation Temperature evolution can be derived from first law of thermodnamics (Ch.4, 4-2): DDDD DDDD = cc pp DDDD 1 DDDD ρρ DDDD DDDD If the heating rate is ero ( DDDD DDDD DDDD = 1 DDDD DDDD ρρcc pp DDDD (Eq. 6-13) = 0, Ch. 4.3.1) then: The temperature of a parcel will decrease/ increase in ascent/ descent with decreasing/increasing pressure. Introduction of potential temperature θθ (Eq. 4-17). θθ = TT pp 0 pp QQ: heat DDDD : diabatic heating rate per unit mass DDDD cc pp : specific heat at constant pressure T: temperature ρρ: densit p: pressure; p 0 = 1000 hpa κ=r/c p R: gas constant for dr air 38

5. Thermodnamic equation Thermodnamic equation - potential temperature Eq. 6-13 for potential temperature θθ (see book for details): DDθθ DDDD = pp pp 0 κ DDDD DDDD cc pp (Eq. 6-14) Note: For adiabatic motions δδδδ = 0 θθ is conserved. 39

3. Equations of motion - nonrotating fluid Coordinate sstems in meteorolog Cartesian coordinates (,,,t) Spherical coordinates (λ,ϕ,,t) Height coordinates: geometrical altitude (), pressure (p) and potential temperature (θ)

Summar 3.-5.) Summar 5 equations for the evolution of the fluid (5 unknowns) Ia.) Ib.) vv Ic.) ww II.) tt + uu + uu vv + uu ww III.) DDDD DDDD = cc pp + vv + w + 1 ρρ vv + vv + w vv + 1 ρρ ww + vv + w ww + 1 + g = F ρρ = F (6-7a) = F (6 7b) (6-7c) + ρρuu = 0 (6-9 or 6-11/6-12) DDDD 1 DDDD ρρ DDDD DDDD (6-13 or 6-14) Restrictions: - application to average motion is often incorrect i.e. turbulence, - fied coordinate sstem. 42

5. Equations of motion - rotating fluid Forces on rotating sphere Fictitious forces: - occur in revolving/accelerating sstem (e.g. car in curve) - occur on the earth s surface (Fied sstem on earth s surface forms an accelerated sstem, because a circular motion is performed once a da. Movement of earth around the sun compared to earth s rotation is insignificant.) Coriolis force Centrifugal force 43

5. Equations of motion - rotating fluid Inertial - rotating frames Marshall and Plumb (2008) 44

5. Equations of motion - rotating fluid Transformation into rotating coordinates Consider figure 6.9: uu iiii = uu rrrrrr + Ω rr (Eq. 6-24) DDAA DDDD iiii = DDAA DDDD rrrrrr + Ω A (Eq. 6-26) Setting A =r and A uu iiii in Eq. 6-24 and 6-26 we can derive: DDuu iiii DDDD iiii = DD DDDD rrrrrr + Ω uu rrrrrr + Ω rr = DDuu rrrrrr DDDD rrrrrr + 2Ω uu rrrrrr + Ω Ω rr (Eq. 6-27) rr : position vector A : an vector Ω: rotation vector DDrr DDDD rrrrrr = uu rrrrrr 45

5. Equations of motion - rotating fluid Rotating equations of motion Substituting DDuu iiii from Eq. 6-27 into Eq. 6-6 (inertial frame DDDD iiii equation of motion ) we derive in rotating frame, dropping subscript rot : DDuu + 1 DDDD ρρ pp + g = 2Ω uu + Ω Ω rr + F (Eq. 6-28) Pressure gradient accel. Gravitational accel. Coriolis acceleration Centrifugal acceleration Friction acceleration 46

Centrifugal force Centripetal force Directed radiall outward. Angular velocit Ω If no other forces are present the particle would accelerate outwards. Fictitious force, balanced b Centripetal force F C = M Ω (Ω r) Centripetal force F Cp Centre r Bod (mass M) Centrifugal force F C F Cp = - F C 47

5. Equations of motion - rotating fluid Centrifugal accel. modified gravitational potential Combine the gradient of Centrifugal potential Ω Ω rr = Ω2 r 2 Gravitational potential g = (g) in Eq. 6-28: 2 and DDuu DDDD + 1 ρρ pp + ΦΦ = 2Ω uu + F Eq. 6 29 Pressure gradient accel. Gravitational +Centrifugal accelerations Coriolis acceleration Friction acceleration ΦΦ = g Ω2 r 2 2 Eq. 6 30 ΦΦ(Greek Phi ) is modified gravitational potential on Earth. 48

5. Equations of motion - rotating fluid On the sphere Shallow atmosphere appro. a+ a: r=(a+)cosφφ a cosφφ, Eq. 6-30 becomes: ΦΦ = g Ω2 aa 2 cos 2 φφ 2 Definition of geopotential surfaces: = + Ω2 aa 2 cos 2 φφ 2g (Eq. 6-40) Ω= 7.27 10-5 s -1 a = 6.37 10 6 m Marshall and Plumb (2008) 49

GEF1100 Autumn 2014 30.09.2014 GEF 1100 Klimasstemet Chapter 6: The equations of fluid motion Prof. Dr. Kirstin Krüger (MetOs, UiO) 50

5. Equations of motion - rotating fluid Coriolis acceleration NH (viewed from above the NP): Ω > 0: rotation anticlockwise SH (viewed from above the SP): Ω < 0: rotation clockwise Absence of other forces: DDuu DDDD = 2ΩΩ uu Eq. 6-31 ΩΩ = 0 Ω Ω = 0 Ω ccccccφ Ω sin φ : Angular velocit Marshall and Plumb (2008) 52

5. Equations of motion - rotating fluid Coriolis force on the sphere Coriolis parameter Assuming that 1) Ω is negligible as Ωu<<g; 2) w<<u h leads to: 2ΩΩ uu ( 2Ω sin φ vv, 2Ω sin φ uu,0) (Eq. 6-41) = ff uu Coriolis parameter ff: ff = 2Ω sin φφ (Eq. 6-42) Note: Vertical component of the Earths rotation rate, which matters. Ω= 7.27 10-5 s -1 a= 6.37 10 6 m Marshall and Plumb (2008) 53

5. Equations of motion - rotating fluid Directional movement Coriolis force In direction of movement, in NH: deflection of air particles to the right SH: Deflection of air particles to the left Schematic picture for Coriolis force takes place. 54

5. Equations of motion - rotating fluid Equations of motion Coriolis parameter DDuu DDDD + 1 ρρ pp + ΦΦ +f uu = F (Eq. 6 43) Fluid in a thin spherical shell on a rotating sphere, appling hdrostatic balance for vertical component and neglecting F compared with gravit: DDu DDDD + 1 ρρ DDvv DDDD + 1 ρρ 1 ρρ pp ffff = F (Eq. 6 44) pp + ffff = F pp + g = 0 55

Take home message 5 equations for the evolution of the fluid with 5 unknowns (u,v,w,p,t): equations of motion (3), conservation of mass (1), thermodnamic equation (1) Equations of motion on a non-rotating fluid: Pressure gradient force, gravitational force and friction force act. DDuu DDDD + 1 ρρ pp + g = F Equations of motion on a rotating fluid: Pressure gradient force, modified gravitational potential (gravitational and centrifugal force), Coriolis force and friction force act. DDuu + 1 pp + ΦΦ +f uu = F DDDD ρρ 56