# 2.1 Introduction To Lagrangian (Material) Derivatives

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22.1The equations governing atmospheric flow.Introduction to Lagrangian (Material) derivativesThe equations governing large scale atmospheric motion will be derived from a Lagrangianperspective i.e. from the perspective in a reference frame moving with the fluid. The Eulerian viewpoint is normally the one we are concerned with i.e. what is the temperatureor wind at a fixed point on the Earth as a function of time. But, the Lagrangian perspective is useful for deriving the equations and understanding why the Eulerian winds andtemperatures evolve in the way they do.The material derivative (D/Dt) is the rate of change of a field following the airparcel. For example, the material derivative of temperature is given byDT T u · T,Dt t(1)where the first term on the RHS is the Eulerian derivative (i.e. the rate of a change at afixed point) and the second term is the advection term i.e. the rate of change associatedwith the movement of the fluid through the background temperature field. If the materialderivative is zero then the field is conserved following the motion and the local rate ofchange is entirely due to advection.2.1.1Material derivatives of line elementsWhen deriving the equations of motion from a lagrangian perspective we will considerfluid elements of fixed mass. But, their volume may change and it is therefore necessaryto consider the material derivative of line elements (vectors joining two points). ConsiderFig. 1 which shows a line element δ r at position r at a time t. This line element couldFigure 1: Schematic of a line element moving with a velocity field.1

Figure 2: The infinitesimal mass element δM .represent, for example, the side of an infinitesimal fluid element. If the line element wereconnecting two fixed points in space then it will not change with time, but, if it connectstwo points that are moving with the velocity field of the fluid it will change its positionand orientation over time. The line element at time t δt is given byδ r(t δt) δ r(t) v ( r δ r)δt v ( r)δt δ r(t δt) δ r(t) v v ( r δ r) v ( r) δ r(2)δt rThe left hand side is just the material derivative of the line element. So we have theimportant result thatDδ r δ r · v(3)Dtr. So, different shear terms can affect the size and orientation of line elements.where v D DtFor a particular dimension q we have the general result that 1 Dδq Dq (4)δq Dt q DtThe equations that govern the atmospheric flow will now be derived from a Lagrangianperspective. These are Mass Conservation, Momentum Conservation and EnergyConservation.2.2Conservation of mass (Equation of continuity)We consider an infinitesimal fluid element of constant mass δM depicted in Fig. 2 whosedensity (ρ) and volume (δV ) may vary with time. Since the mass of the fluid element isconstant then DδM/Dt 0, where δM ρδV . So,DρDδVDδM 0 δV ρ 0DtDtDt2(5)

Consider first the rate of change of the infinitesimal volumeDδVDDδxDδyDδz (δxδyδz) δyδz δxδz δxδy(6)DtDtDtDtDtBut, from the material derivative of line elements (Eq. 3) this can be written Dδx Dδy Dδz u v wDδV δxδyδz δV ( . v )δV.Dt x Dt y Dt z Dz x y z(7)So,DδV ( . v )δV(8)DtSo, the fractional rate of change of the volume is equal to the divergence of the velocityfield. So, back to the eqution for mass coninuity we haveDδVDρDρDρδV ρ 0 δV ρ( . v )δV 0 ρ · v 0DtDtDtDtfor arbitrary δV . This can also be written in flux form as ρ · ( v ρ) 0 tTo summarise, mass continuity is given byDρ ρ ρ · v · ( v ρ) 0Dt t(9)(10)(11)Thus, a fluid element in a divergent flow field will experience a reduction in density.2.3Conservation of momentumHere, we are again considering our infinitesimal air parcel of constant mass (δM ) withvariable volume (δV ) and density ρ. We want to know how the momentum of the parcelwill change. The rate of change of momentum will be equal to the sum of the forces actingon the parcel.D(δM v )D v δM(12)DtDtThere are several forces we may have to consider when examining the motion of fluidelements in the atmosphere or ocean:F Pressure - Consider the vector area element whose normal vector is given by d s inFig. 2. The pressure acting on the area element is dF p pd s. Integrating over theRsurface area of Rthe element gives F p pd s which from the divergence theoremcan be written pdV pδV as δV 0. Gravity - The gravitational force acting on the fluid element in geometric heightcoordinates is Fg δM g which is often written as Fg δM Φ where Φ gz isthe Geopotential and g is the acceleration due to gravity which for many purposescan be taken to be a constant. The reason for writing it in this form is that we mayconsider flow in different height coordinates (commonly pressure is used). Thus aparticular vertical level in our coordinate system may not necessarily be horizontaland there may ba a component of gravity acting along that level.3

Figure 3: (a) Schematic illustrating the coriolis force as conservation of angular momentum,(b) schematic illustrating the tangent plane approximation. Viscosity - Viscosity is a measure of the resistance of a fluid. The viscosity arisesfrom shear stress between layers of a fluid that are moving with different velocities.The viscous force per unit mass is often written as F ν ν 2 v where ν is thekinematic viscosity. For the atmosphere below around 100km the viscosity is sosmall that it is normally negligible except for within a few cm at the Earth’s surfacewhere the vertical shear is very large. (See Holton 1.4.3 for more on viscosity). TheReynolds number is a measure of the importance of viscosity. It is the ratio ofinertial forces to viscosity. In most of the atmosphere the Reynolds number is largeand viscosity is relatively unimportant.So, our rate of change of momentum in Eq. 12 is given by the sum of these forces.Dividing through by δM gives us an expression for the acceleration of the fluid parcel inan inertial reference frame. pD v Φ ν 2 vDtρ(13)If the fluid is at rest then we have hydrostatic balance 2.3.1 p p Φ 0 k ρg kρ z(14)Adding in rotationIn the above section we related the rate of change of momentum of our fluid element tothe sum of the forces acting on the fluid element. But, that only applies when we areconsidering motion in an inertial reference frame. Normally when examining atmosphericmotion we are not looking at motion in an inertial reference frame. Rather we are concerned with how the atmosphere is moving with respect to the Earth’s surface i.e. in arotating reference frame. We therefore have to take this into account when determiningmomentum balance.For example, consider a parcel of air which is at a fixed position with respect to the The position of the fluidEarth’s surface. The Earth rotates with an angular velocity Ω.4

element does not alter in the rotating reference frame of the Earth. But, from the pointof a view of an observed in the Inertial reference frame the position vector will move asit rotates with the Earth. The rate of change of the position vector as viewed from theinertial reference frame is given by (See Holton 2.1.1) D r r Ω(15)Dt IIf the air parcel now has a velocity with respect to the fixed surface of the Earth denotedby (D r/Dt)R , then an observed in the inertial reference frame will see the air parcelmoving with a velocity D rD r r(16) ΩDt IDt Ri.e. the sum of the velocity in the rotating reference frame and the velocity associatedwith the movement of the reference frame. vI vr (Ω r)(17)If we now consider the rate of change of the inertial velocity in the inertial reference frame:it will be related to the rate of change of the inertial velocity in the rotating referenceframe by the same transformation (16) i.e. D vID vI vI . Ω(18)Dt IDt RBut, combining with (17) it can be shown that D vID vR vR Ω (Ω r) 2ΩDt IDt R(19)where the 2nd and 3rd terms on the RHS are the Coriolis force and the centrifugal forcerespectively. These are ficticious forces introduced by the fact that we are examining themotion in a non-inertial reference frame. (Ω r ) where r is the The centrifugal force - The can be rewritten as Ωcomponent of the position vector that is perpendicular to the axis of rotation. This r )Ω (Ω. Ω). r where the first term on the RHS is zerocan can be re-written as (Ω. 2giving Ω r . This can then be written as the gradient of a scalar potential ( ΦCE )where ΦCE Ω2 r /2. Therefore, the centrifugal force is often combined with thegravitational force in (13) but for many purposes the centrifugal force is small andcan actually be neglected. Coriolis force - The coriolis force can be thought of in terms of angular momentumconservation. For examine, consider an air parcel starting at position (1) at lowlatitudes in the NH in Fig. 3 (a). Suppose now a force acts to shift that air parcelpoleward to position (2). No torque has acted on the parcel in the zonal directionand so the angular momentum of the air parcel around the Earths axis of rotationmust be conserved i.e. Ω1 r12 Ω2 r22 , but r1 r2 Ω1 Ω2 . So, as the airparcel moves poleward, it’s angular velocity in the zonal direction increases. So, anobserver at position (1) rotating with the Earth would see a trajectory of the airparcel that looks like it is being deflected to the right.5

So, in (13) the acceleration of the air parcel is related to the forces that are acting onit. But, that applies for motion in an inertial reference frame.What we normally care D vRabout is the motion in a rotating reference frame i.e. Dt R . We can therefore combine(13) and (19) to give D vR vR p Φ ν 2 v 2Ω(20)Dt Rρwhere we have either neglected the coriolid force or lumped it in with the gravitationalpotential.From now on we shall drop the subscript R and assume we are examining motion inthe rotating reference frame of the Earth. So, to conclude, momentum balance in therotating reference frame of the Earth where pressure, gravity, viscous and possibly otherfrictional forces are acting is given byD v v p Φ ν 2 v 2ΩDtρ2.3.2( f riction)(21)Scale analysis of momentum balanceConsidering the typical horizontal velocity (U ) and typical length scales (L) of the systemwe can perform scale analysis on the the first two terms of Eq. 39. The first termUU2D v DtTAL(22)where we have made use of the fact that the relevant timescale for the material derivativeis the advective timescale. The second term scales as2Ω v 2Ω v sinφ f U(23)where f is the coriolis parameter. The angle φ is the angle between the rotation vectorand the velocity vector which when considering the horizontal velocity components onthe Earth’s surface is simply the latitude (See Fig. 3 (b)). Thus 1./f is the relevanttimescale for the effects of the Earth’s rotation as was discussed in Section 1. Takingthe ratio of the first term to the second term we end up with the ratio U/f L which isthe Rossby Number Ro . So, the Rossby number is a dimensionless parameter that comesfrom scale analysis of momentum balance. It is the ratio of the intrinsic acceleration tothe acceleration associated with the coriolis force. If the Rossby number is small thenthe effects of the Earths rotation are relatively important and vice-versa. If we are in theregime of small Rossby number then if you apply a force on an air parcel and you’re in therotating reference frame of the Earth then you’re not going to see the parcel acceleratingin the direction of the force because there will be an acceleration due to the coriolis force.We see from the terms involved in the Rossby number that the coriolis force is relativelyimportant if we are considering motion that is slow and/or large scale. This is the typeof motion that we will be concerned with.2.4Conservation of Energy (Thermodynamic Equation)Again we consider our air parcel of fixed mass δM . We define I the internal energy perunit mass, η the entropy per unit mass and α the volume per unit mass. Thus the total6

internal energy is given by δU IδM , the total entropy is given by δS ηδM and thetotal volume is given by δV αδM . We take the first law of thermodynamics as ourstarting point for deriving the thermodynamic equation for atmospheric flow. The firstlaw of thermodynamics may be written asdU T dS pdV(24)where d represents the change in our thermodynamic variable. That is the change ininternal energy of our mass element is given by the difference between the heat inputinto the mass element and the work done by the mass element. This is in terms of theextensive quantities so it depends on the size of the system. But, we can write it in termsof the quantities per unit mass (or intensive variables) as followsd(IδM ) T d(ηδM ) pd(αδM )(25)So, over some time interval δt we haveT d(ηδM ) pd(αδM )d(IδM ) δtδtδt(26)As the time interval δt tends to zero these tend to the rate of change following the parcel(the material derivative) soD(IδM )D(ηδM )D(αδM ) T pDtDtDt(27)But, δM is conserved following the motion.DηDαDI T pDtDtDt(28)This is our first law of thermodynamics in terms of the entropy per unit mass and thevolume per unit mass. These are rather difficult quantities to measure, but matters may besimplified when we are considering motion of an ideal gas, which we are for most purposesin atmospheric dynamics. It is much easier if we can use pressure (p) and temperature(T ) as our thermodynamic variables as these are quantities we can measure. For an idealgas we have several useful relationshipspα RTI CV TCp CV R(29)where R is the gas constant, CV is the specific heat capacity at constant volume and CPis the specific heat capacity at constant pressure. So, we can rewrite 28 asCVDTDηDTRT Dp T R DtDtDtp Dt(30)which after some rearranging can be writtenTDTRT DpDη Cp DtDtp Dt(31)This is an expression for the rate of change of entropy as a function of temperature andpressure. But, if we are considering adiabatic motion of an ideal gas then entropy should7

be conserved. This leads us to the definition of potential temperature. For adiabaticmotion of an ideal gas we have Dln pTκCp DTR DpDlnpκDηDlnT CP Cp 0(32)DtT Dtp DtDtDtDtwhere κ R/Cp . Therefore this ratio T /pκ must be a constant since the Cp is a constantand non-zero. If we have an air parcel that has a temperature T at a pressure p then ifit is brought adiabatically to a reference pressure po then it will have a temperature (θ)at that reference pressure given by κθpoT(33) θ T pκopκpThis quantity θ is known as the potential temperature. It is the temperature that anair parcel would have if it were brought adiabatically to the reference pressure po . Thereference pressure is normally taken to be the surface pressure ( 1000hPa). Potentialtemperature increase upward in the troposphere even though temperature decreases because the pressure is also decreasing. A positive potential temperature gradient impliedstability as given by the Brunt-Vaisala frequency in section 1.By comparison of 33 and 32 it can be seen that the rate of change of entropy is relatedto the potential temperature byDηDlnθ Cp(34)DtDtand thusη Cp lnθ(35)Surface of constant potential temperature are therefore surface of constant entropy (isentropic surfaces). We can see from Eqs. 32 and 33 that potential temperature must beconserved following adiabatic motion of an ideal gas. We can therefore write our thermodynamic equation asDθ 0(36)DtIf there are sources of diabatic heating then we can make use of 34 and the relationshipbetween the heat input and the change in entropy (dQ T dη) to write the thermodynamicequation in the presence of diabatic effects asθDθ Q̇(37)DtTNote: the above derivation of the thermodynamic equation relied on relationships thatare valid for an ideal gas. In the atmosphere what we are really concerned with is themotion of an ideal gas although there may need to be modification to include effects likemoisture (see Vallis chapter 1). For, liquids a different equation of state is necessary andthe thermodynamic equation takes a different form.Cp2.5Summary of the primitive equationsThe above equations that we have derived for atmospheric flow are often known as theprimitive equations. The primative equations are summarised as follows ρ .( v ρ) 0 tCONTINUITY8(38)

D v p 2Ω v Φ ν 2 vMOMENTUM(39)DtρθDθDθ 0orCp Q̇THERMODYNAMIC(40)DtDtTBut, here we have 6 variables (ρ, u, v, w, p, θ) and only 5 equations. We need the diagnosticequation of state which, for an ideal gas is given bypα RT(41)We will be using these equations to look at simple cases in the atmosphere and understandwhy certain aspects of the large scale circulation behave the way they do.2.6Geostrophic BalanceMomentum balance in the absence of viscosity and friction is given byD v v p Φ. 2ΩDtρ(42)The assumptions we make about the characterisctics of large scale extra-tropical motion inthe atmosphere allow us to identify the dominant balances for such motions. We considerflow for which the Rossby number is small i.e. the first term in 42 is much smaller thanthe second. In geometric height coordinates we may neglect the component of gravityacting in the horizontal direction. Thus, in the horizontal we have the dominant balanceoccuring between the second and third terms (the coriolis force and the horizontal pressuregradient). This gives 1 p p(ug , vg ) ,(43)ρf y xThe above expression gives the Geostrophic wind: the horizontal wind for which thecoriolis force exactly balances the horizontal pressure gradients. This balance is knownas Geostrophic balance. Geostrophic balance is often a reasonable approximation forlarge scale motion away from the equator. Examination of Eq. 43 shows that, in the NHthe flow will be clockwise around high pressure (anticyclonic) systems whereas the flowwill be anti-clockwise around low pressure (cyclonic) systems. The opposite is true in theSH as the coriolis parameter (f ) is of opposite sign.2.7Hydrostatic balanceIn the velocity was zero it can be seen from Eq. 42 that there would be a balance betweenthe vertical pressure gradient and the gravitational force - Hydrostatic balance. But,infact, if we make the assumption that we are examining flow of small Rossby numberand small aspect ratio then the supplementary handout demonstrates that even in thepresence of a non-zero velocity the dominant balance in the vertical is between the verticalpressure gradient and the gravitational force. So, for large scale flow hydrostatic balanceis the dominant balance in the vertical.1 p Φ ρ z zor9 p ρg z(44)

Eq (44) together with the ideal gas law gives an expression for the geopotential of apressure level (known as the Hypsometric equation)Z sΦp Φs (45)RT dlnp0 ,pwhere Φp is the geopotetial of a particular pressure level and Φs is the geopotential of thesurface. The geopotential height (Z) is related to the geopotential via Z Φ/g and soEq. 45 gives an expression for the geopotential height of a particular pressure level. Itis related to the integral of temperature below that pressure level and is approximatelyequal to the geometric height in the troposphere and lower stratosphere.2.7.1Using pressure as a vertical coordinateFrom hydrostatic balance we have a monotonic relationship between pressure and geometric height and so pressure is an equally valid vertical coordinate. It is often advantageousto formulate the primitive equations in pressure coordinates as we lose the dependenceon density. Mass conservation becomes u x p v y p ω 0 p(46)where ω Dp/Dt is the pressure velocity. So, for a hydrostatic system in pressurecoordinates the velocity field is non-divergent. The horizontal component of inviscidmomentum balance in the absence of friction in pressure coordinates becomesD vH vH Φ 2Ω(47)Dtwhere vH represents only the horizontal velocity (u, v) and our geopotenti

The equations governing large scale atmospheric motion will be derived from a Lagrangian perspective i.e. from the perspective in a reference frame moving with the ﬂuid. The Eu-lerian viewpoint is normally the one we are concerned with i.e. what is the temperature

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