Dynamic Meteorology Lecture Notes PDF

Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Chapter-I Circulation and Vorticity Circulation: Definition: Circulation...

Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Chapter-I Circulation and Vorticity Circulation: Definition: Circulation is defined as a macro-scale measure of rotation of fluid. Mathematically it is defined as a line integral of the velocity vector around a closed path, about which the circulation is measured. r Circulation may be defined for an arbitrary vector field, say, B. Circulation ‘ C B ’ of r an arbitrary vector field B around a closed path, is mathematically expressed as r r r a line integral of B around that closed path, i.e., C B = ∫ B. d l. In Meteorology, by the term, ‘Circulation’ we understand the circulation of velocity vector. Hence, in Meteorology circulation around a closed path is given by C = ∫V. dl....(C1.1). From this expression it is clear that circulation is a scalar quantity. Conventionally, sign of circulation is taken as positive (or negative) for an anticlockwise rotation (or for a clockwise rotation) in the Northern hemisphere. Sign convention is just opposite in the Southern hemisphere. Since we talk about absolute and relative motion, hence we can talk about absolute circulation and relative circulation. They are respectively denoted by C a and C r respectively and are defined as follows: Ca = ∫V a. dl …. (C1.2) and Cr = ∫V r. dl …. (C1.3) Where V a and V r are the absolute and relative velocities respectively. 1 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Stokes Theorem:- r It states that the line integral of any vector B around a closed path is equal to the surface integral of ∇ × B.n over the surface ‘S’ enclosed by the closed path, where n̂ is the outward drawn unit normal vector to the surface ‘S’. r So, ∫ B ⋅ dl = ∫∫ (∇ × B) ⋅ nˆ ds. The Circulation Theorems: Circulation theorems deal with the change in circulation and its cause(s). r For an arbitrary vector field, B the circulation theorem states that the time r rate of change of circulation of B is equal to the circulation of the time rate of r change of B , i.e., r d r dB r dt ∫ B. d l = ∫. dl …………(C1.4) dt r This theorem may be applied to the absolute velocity vector ( Va ) as well r as to the relative velocity vector ( Vr ). Kelvin’s Circulation theorem: It is the circulation theorem, when applied to the absolute velocity r ( Va ) of fluid motion. So according to Kelvin’s Circulation theorem, d aCa d a Va dt = ∫ dt ⋅ dl …..(C1.5). Proof: We know that C a = ∫V a. dl d aCa d a r r dt ∫ So, = V a. d l dt r d aCa d aV a r r d r Or, = ∫. dl + ∫ Va. a (dl ) dt dt dt r d C dV r r r Or, a a = ∫ a a. dl + ∫ Va. d aVa dt dt r r r d aCa d aV a r Va.Va Or, = ∫. dl + ∫ d a ( ) dt dt 2 2 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. r d C d aV a r Or, a a = dt ∫ dt. dl , as the line integral of an exact differential around a closed path vanishes. d aCa dC r Conventionally, or are known as acceleration of circulation dt dt (absolute or relative). So, in Meteorology, circulation theorem simply states that the acceleration of circulation is equal to the circulation of acceleration. A corollary to Kelvin’s circulation theorem: We know that equation for absolute motion is given by, r d aV a 1 r r r = − ∇p + g ∗ + F ……….(C1.6), where symbols carry their usual dt ρ significances. r r ⎛r ⎞ GM Here, g ∗ = − 2 ⎜ ⎟ is the gravitational attraction exerted by earth on a ⎝r⎠ r r unit mass with position vector, r , with respect to the centre of the earth. It is clear r that g ∗ is a single valued function of ‘ r ’. Also it is known that all force fields which are single valued functions of distance ( r ), are conservative field of forces. r (‘Dynamics of a particle’, by S.L.Lony). Hence, g ∗ is a conservative force field. It is also known that work done by a conservative force field around a closed path is zero. r∗ r Hence, ∫g. dl = 0 ….(C1.7). r Again, from Stoke’s law we know that for a vector field, B , r r r r ∫ B. dl = ∫∫ ∇ × B. nˆ ds ……….(C1.8) s Where S is the surface area enclosed by a closed curve, around which the r circulation of B is measured, and ‘ n̂ ’ is the outward drawn unit vector normal to the surface area S. r r 1 r r r ⎛ 1 r ⎞ ∇ρ × ∇p So, ∫ − ∇p. dl = ∫∫ ∇ × ⎜⎜ − ∇p ⎟⎟. nˆ ds = ∫∫. nˆ ds ……….(C1.9) ρ s ⎝ ρ ⎠ s ρ 2 3 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Hence, using (C1.6), (C1.7) and (C1.9) in (C1.5), we have for friction less flow, r r d aCa ∇ρ × ∇p = ∫∫. nˆ ds …………..(C1.10) dt s ρ2 We know that in a barotropic atmosphere the density, ρ , is a function of pressure only, i.e., ρ can be expressed as, ρ = f ( p). r r r r r r Hence, ∇ρ = f ′( p)∇p ⇒ ∇ρ × ∇p = 0 , where, 0 is null vector. d aCa Therefore, for a frictionless barotropic flow, =0…….(C1.11). This is a dt direct corollary to the Kelvin’s theorem. Hence from Kelvin’s circulation theorem it may be stated that for frictionless flow change in absolute circulation is solely due to the baroclinicity of the atmosphere. Solenoidal vector and Jacobian: Suppose, A, B are two scalar functions. Then, Jacobian of these functions, is denoted by J ( A, B) and is given by, ∂A ∂A ∂y ˆ r ∂x r r r J ( A, B) = = k. ∇A × ∇B. Also, ∇A × ∇B is called A, B Solenoidal ∂B ∂B ∂x ∂y r vector and is denoted by, N A, B. So, the vertical component of solenoidal vector is the Jacobian. Now, it will be shown that, J ( A, B) represents change in A( x, y ) along the isolines of B ( x, y ) and vice-versa. r r r r We have, J ( A, B) = ∇A × ∇B = ∇A ∇B sin θ , where,θ is the angle between r r r r ∇A and ∇B. We know that ∇A , ∇B are normal to the isolines of A, B respectively. Hence the angle between isolines of A, B is also θ. If α is the r r r angle between isolines of B and ∇A , then θ = 900 - α. So, J ( A, B) = ∇A ∇B cos α. r r Now, ∇A cos α represents the magnitude of the projection of ∇A on the isoline 4 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. r r of B. As ∇A represents change of A , hence it follows that ∇A cos α represents the change of A along the isolines of B. Thus, for a given gradient of B , J ( A, B) represents the change of A along the isolines of B. Similarly, it can be shown that for a given gradient of A , J ( A, B) represents the change of B along the isolines of A. The above has been shown in figure 1.1. From this figure it is clear that as the magnitude of α increases, the magnitude of the change in A (Or B ) along the isolines of B (Or A ) increases. Hence, the magnitude of the Jacobian increases as the angle between the isolines decreases. It is maximum when θ = 0 0 and is zero when θ = 90 0. Barotropic and Baroclinic Atmosphere: Here we shall discuss the salient features of the solenoid vector. Solenoid vector, denoted by N ρ , P or N T , p is given by N ρ , P = ∇ρ × ∇ p …. (C1.12) or r N T , p = ∇T × ∇ p …. (C1.13). When the atmosphere is barotropic, then, there is no horizontal r temperature gradient. Hence in such an atmosphere, ∇T = 0̂ [ 0̂ is the null vector]. ( R r ) r Hence in such an atmosphere, ∇ p ×∇T = 0̂. p N T , p = 0̂. Hence ∇T ll ∇P. Hence in such case, the isobars and isotherms (or the isolines of density ρ ) are parallel to each other. This has been shown in fig.1.2. But if the atmosphere is not barotropic, then these lines are no longer parallel, rather they intersect each other. Now, when they intersect, they form small rectangles like ABCD (shown in fig 1.3). Such rectangles are called solenoid. It is shown below that the magnitude of Solenoid Vector is equal to the number of solenoids formed in unit area in the vertical plane. 5 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Area of a single solenoid ABCD = a h1 , where a is the length of the side AB and h1 is the length of the altitude DE, as shown in figure 1.3. Now, h1 = bsin θ , where, b is the length of the side AD and θ is the angle between the sides AB and AD. Hence, area of the solenoid ABCD = absin θ. r r r r 1 1 1 Now, ∇T × ∇p = ∇T ∇p sin θ = sin θ = , where, h2 is the length h2 h1 a b sin θ of the altitude BF. So, area absin θ is contained in 1 solenoid. 1 Hence, unit (= 1) area is contained in numbers of solenoid. So, a b sin θ the magnitude of above solenoidal vector represents the number of solenoids in unit area in a vertical plane. Practically the angle between isobar and isotherms gives a qualitatively measure of baroclinicity of the atmosphere. Because as the angles are smaller, the isobars and isotherms are very close to be parallel to each other i.e. the atmosphere is mostly barotropic. But as the angle increases, the isotherms and isobars become far away from being parallel to each other i.e. the atmosphere is mostly baraclinic. Also it is worth to note that as the angles between isotherms and isobars are smaller, numbers of solenoids are also smaller and if angle increases, the numbers of solenoids are also increases. These have been shown in figures (1.4 & 1.5).From the figures 1.4 and 1.5 we can see how the increase in angle between isobars and isotherms can lead to increase in change in T along the isobars. So in the day to day charts to examine the qualitative measure of baroclinicity we need to estimate only the angle between isobars and isotherms or in the constant pressure chart we need to examine the angle between contour lines and the isotherms. 6 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Bjerknees Circulation Theorem: Kelvin’s circulation theorem tells us about the change of absolute Circulation. But it is more important to know about the change of circulation with respect to the earth. Hence it is more important to know the change of relative circulation. Bjerkness circulation theorem tells us about the change in relative circulation According to Bjerkness circulation theorem, we have dC r dC a dS = -2 Ω E ………………….. (C1.14) dt dt dt r r r r Proof: We know that, Va = V + Ω × r. ( ) r r r r r r r ⇒ ∫ Va.dl = ∫ V. dl + ∫ Ω × r. dl r r r ⇒ C a = C r + ∫∫ ∇ × (Ω × r ). nˆ ds (Stoke’s theorem used for 2nd line integral) S r ⇒ C a = C r + ∫∫ 2Ω. nˆ ds S r r r Now, Ω. nˆ = Ω nˆ cos (Ω, nˆ ) = Ω sin φ , where, φ is the latitude of the area r element ds and Ω = Ω. Hence, ⇒ C a = C r + 2Ω ∫∫ dsSinφ = 2ΩS E S Where, S E = ∫ dS E = ∫ dS Sin φ and ds Sin φ is the area of the projection of ds on the equatorial plane. dC a The first term , have already been discussed in the Kelvin’s circulation dt dS E theorem. Now we shall discuss the 2nd term -2 Ω. dt Considering the effect of the 2nd term independently the Bjerkness circulation theorem gives us Cr 2 − Cr1 = −2Ω( S 2 Sinφ2 − S1Sinφ1 ) …………………………(C1.15) Where Cr1 = Initial relative circulation; 7 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Cr 2 = Final relative circulation S1 = Initial area enclosed by the closed path S2 = Final area enclosed by the closed path φ1 = Initial Latitude φ2 = Final Latitude Thus the above equation tells us that the change in relative circulation may be due to (i) change in area enclosed by the closed path (ii) change in latitude (iii) Non uniform vertical motion superimposed on the circulation Effect of the change in area enclosed by the closed path on the change in relative circulation : If the area ‘S’ enclosed by the closed path increased from S1 to S2 , remaining at the same latitude ’ φ ’, then the resulting change in relative circulation is given by C r 2 − C r1 = −2ΩSinφ ( S 2 − S1 ) < 0 , since, S2 > S1. Thus Cyclonic circulation decreases as the area enclosed by the closed circulation increases. Physically it may be interpreted as follows: Area enclosed by a closed circulation increases if and only if the divergence increases or convergence decreases. Then due to the Coriolis force the stream line turn anti-cyclonically or the already cyclonically turned streamlines turn less cyclonically. As a result of which cyclonic circulation reduces. Similarly due to convergence when the area enclosed by the circulation decreases, the cyclonic circulation increases. Effect of the change in latitude on the change in relative circulation : Now suppose a circulation moves from a lower latitude φ1 to a higher latitude φ2 , without any change in the area enclosed by the circulation. Then the resulting change in the relative circulation is given by Cr 2 − Cr1 = −2ΩS ( Sinφ2 − Sinφ1 ) < 0 8 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Since Sinφ2 > Sinφ1 Hence a circulation loss its cyclonic circulation as it moves towards higher latitude. Similarly it can be shown that when a cyclonic circulation moves towards lower latitude, then it gains cyclonic circulation. Effect of imposition of non uniform vertical motion on the change in relative circulation. Now consider a different situation, when neither the area enclosed by the circulation changes nor the cyclonic circulation moves, but non uniform vertical motion is applied to the closed circulation. Then the inclination of the plane of rotation of circulation with the equatorial plane changes, (shown in figure 1.6) as a result of which SE changes which leads to a change in Cr. This effect is known as TIPPING EFFECT. A possible explanation of sea/land breeze and thermally direct circulation using Kelvin’s circulation Theorem: Sea breeze takes place during day time when ocean is comparatively cooler than land. Hence temperature increases towards land and also we know that temperature decreases upward. (i.e. increased r downward). Thus the temperature gradient ∇ T is directed downward to the land. For the shake of simplicity we assume that pressure over land and sea is r same, but it increases downward. Hence pressure gradient ∇ p is directed r r downward. as shown in figure 1.7. Hence ∇p × ∇T gives the circulation in the r r direction from ∇ p to ∇ T. Also the change in circulation pattern is given by r r ∇p × ∇T. Hence if initially there was no circulation, then the above mentioned r pressure and temperature pattern will generate a circulation directed from ∇ p to r ∇ T , which gives low level flow from ocean to land and in the upper level from land to ocean. This is nothing but sea breeze. Similarly land breeze and any thermally driven circulation pattern may be explained qualitatively. 9 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. VORTICITY: Vorticity is a micro scale measure of rotation. It is a vector quantity. Direction of this vector quantity is determined by the direction of movement of a fluid, when it is being rotated in a plane. Observation shows that when a fluid is being rotated in a plane, then there is a tendency of fluid movement in a direction normal to the plane of rotation (towards outward normal if rotated anti clockwise or towards inward normal if rotated clockwise). Thus due to rotation in the XY plane (Horizontal plane) fluid tends to move in the k̂ direction (i.e. vertical), due to rotation in the YZ plane (meridional vertical plane)fluid tends to move in the iˆ direction (East West) and due to rotation in ZX plane (zonal vertical plane) fluid tends to move in the ĵ direction (N-S). Thus vorticity has three components. Mathematically it is expressed as ∇ × V = iˆξ + ˆjη + kˆζ ….(C1.16), ∂w ∂v ∂u ∂w ∂v ∂u where, ξ = − ;η = − ;ζ = −....(C1.17). ∂y ∂z ∂z ∂x ∂x ∂y In Meteorology, we are concerned about weather, which is due mainly to vertical motion and also only the rotation in the horizontal plane can give rise to vertical motion. So, in Meteorology, by the term vorticity, only the k̂ component of the vorticity vector is understood. Hence, throughout our study only k̂ component is implied by vorticity. ∂v ∂u Thus, hence forth, vorticity = ζ = − ….(C1.18). ∂x ∂y Relation between circulation and vorticity: We know that circulation and vorticity both are measures of rotation. Hence it’s natural that there must be some relation between them. We r r know that, circulation is given by, C = ∫ v. d l Hence, using Stokes theorem we have, C = ∫ v ⋅ dl = ∫∫ ∇ × v ⋅ k ⋅ ds = ∫∫ ζds (As in the present study, rotation is in the horizontal plane, hence, nˆ = kˆ ).. 10 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. dc Hence, = ζ …..(C1.19).Thus, vorticity is the circulation per unit area. ds Vorticity for solid body rotation Let us consider a circular disc, of radius ‘a’, rotating with a constant angular velocity ω about an axis passing through the centre of the disc, as shown in figure 1.8 r r Then the circulation of the disc = ∫ v ⋅ d l = c (say) Now tangential component of v = ω a And dl = adθ 2π 2π ∴ c = ∫ a 2ω dθ = a 2ω ∫ dθ = 2π a 2ω 0 0 2π a 2 w Vorticity= Circulation/Area = = 2 ω ….(C1.20) π a2 Thus for Solid body rotation, the vorticity is twice the angular velocity i.e. 2 ω. Relative vorcity and the Planetary Vorticity ∂v ∂u Relative vorticity = K ⋅ ∇ × Vr = ζ = − ∂x ∂y To understand the Planetary Vorticity, we consider an object placed at some latitude on the earth’s surface. Consider the meridional circle passing through the object shown in figure 1.9 Then as the Earth rotates about its axis, the object executes a circular motion (dashed circle in the fig) with radius a cos φ. Now the Circular motion executed by the object is analogous to the solid body rotation. Hence the vorticity of the object = 2 x local vertical component of angular velocity = 2ΩSinφ = f. Now this vorticity is solely due to the rotation of the planet earth. Hence it is known as planetary vorticity. It is to be noted that it is also the coriolis parameter. 11 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Sum of relative vorticity and planetary vorticity is known as absolute vorticity and is denoted by ‘ ζ a ’ Hence ζ a = ζ + f ……(C1.21) Relative vorticity in natural co-ordinate: r ⎛ ∂ ∂ ˆ ∂⎞ In natural co-ordinate (s,n,z), we know ∇ ≡ ⎜ tˆ + nˆ +k ⎟, ⎝ ∂s ∂n ∂z ⎠ where, tˆ, nˆ , kˆ are unit tangent, unit normal and unit vertical vector respectively. Hence, the relative vorticity is given by ⎛ ∂ ∂ ˆ ∂⎞ ˆ ∂v ζ = kˆ.⎜ tˆ + nˆ + k ⎟ × vt = vK s − ……(C1.22) ⎝ ∂s ∂n ∂z ⎠ ∂n Where, v is the tangential wind speed, K s is the streamline curvature and ∂v is the horizontal wind shear across the stream line. The first term vK s of the ∂n ∂v above expression is known as curvature vorticity and the second term − is ∂n known as Shear vorticity. Potential vorticity To understand the concept of Potential vorticity, first we may refer to the popular circus play, where a girl is standing at the centre of a rotating disc. As the girl stretches her arm, the disc rotates at a slower rate and as she withdraws her arms the disc rotates at a faster rate. Generally this example is referred in solid rotation to illustrate the conservation of angular momentum. This example hints us to search a quantity in the fluid rotation, which is analogous to the angular momentum in solid rotation. For that we consider an air column of unit radius. Now, consider that the air column shrinks down i.e. its depth decreases. As it shrinks down, its radius increases and then as per the above example column will rotate at a slower speed. Also if the air column stretches vertically i.e. if its depth increases, then its radius decreases and rate of rotation increases 12 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. So, it’s clear that the rate of rotation of the air column increases or decreases as its depth increases or decreases. Thus for a rotating air column, we can say that the rate of rotation is proportional to the depth of the air column. Now for fluid motion rate of rotation and vorticity are analogous ∴ Vorticity ∝ Depth Vorticity/Depth = constant Thus in the fluid rotation the quantity (Vorticity/Depth) remains constant as in the solid rotation angular momentum remains constant. So this quantity is analogous to the angular momentum. It is known as potential vorticity. Therefore, Potential vorticity of an air column Absolute vorticity ζ+f = = ……(C1.23) Depth h THE VORTICITY EQUATION: This equation tells us about change in vorticity and the possible mechanisms for vorticity production or destruction. This equation is derived from the equation of horizontal motion. Horizontal equation of motion may be re-written as r r ∂VH r 1 r r ∂VH r = −∇ H K H − ∇ H p − (ζ + f )k × VH − w ˆ + F …….(C1.24) ∂t ρ ∂z r Performing ( kˆ. ∇ H × ) on both sides of (C1.24), we obtain, r r r ∂ς ˆ ∇ H ρ × ∇ H p ˆ r ∂t = k. ρ 2 [ ˆ r ˆ] r − k. ∇ H × (ς + f )k × VH − k. ∇ H × w ∂VH ˆ r ∂z r + k. ∇ H × F To simplify the 2nd and 3rd terms on the RHS of above equation, we use the following two vector identity r r r r r r rr r r r r rr r ∇ H × (a × b ) = (∇ H.b )a − (b.∇ H )a − (∇ H.a )b + (a.∇ H )b ( ) ( ) r r r r r r and, ∇ H × (λ a ) = ∇ H λ × a + λ ∇ H × a Hence the 2nd and 3rd terms are respectively 13 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. r r − DH (ς + f ) − (VH. ∇ H )(ς + f ) ,and r r r r ⎛ r ∂V ⎞ r ∂V ⎛ r ∂V ⎞ ∂ ˆr r ⎛ r ∂V ⎞ ∂ς kˆ. ⎜⎜ ∇ H w × H ⎟ + wkˆ.∇ H × ⎟ H = kˆ. ⎜⎜ ∇ H w × H ⎟ + w ( k.∇ H × V H ) = kˆ. ⎜ ∇ H w × H ⎟ ⎜ ⎟+w ⎟ ⎝ ∂z ⎠ ∂z ⎝ ∂z ⎠ ∂z ⎝ ∂z ⎠ ∂z r r respectively, where, DH = ∇ H.VH. Hence, the vorticity equation may be written as d 1 ⎛ ∂ρ ∂p ∂ρ ∂p ⎞ ⎛ ∂w ∂v ∂w ∂u ⎞ ⎛ ∂Fy ∂Fx ⎞ (ζ + f ) = − DH (ζ + f ) + 2 ⎜⎜ ⋅ − ⎟−⎜ − ⎟+⎜ − ⎟⎟ …..(C1.25) dt ρ ⎝ ∂x ∂y ∂y ∂x ⎟⎠ ⎜⎝ ∂x ∂z ∂y ∂z ⎟⎠ ⎜⎝ ∂x ∂y ⎠ The term on the LHS indicates the production/destruction of absolute vorticity and the terms on the RHS indicates possible mechanisms responsible for that. The terms on the RHS are respectively called 1) Divergence term (2) Solenoidal term 2) Tilting term and (4) Frictional term Divergence term:- This term explains the effect of divergence/convergence on the production/destruction of vorticity. If there is divergence then, DH > 0. Hence considering only the effect of this term we have, d (ς + f ) < 0 ⇒ (ς + f ) , The absolute vorticity decreases with time. dt Thus divergence cause cyclonic vorticity to decrease or anti cyclonic vorticity to increase. This can be explained physically also. Due to divergence, the stream line turns anti cyclonically or cyclonic turning, exists already, decreases by the effect of Coriolis force. It is shown in figure 1.10. Similarly, it can be shown that due to convergence [when D < 0] ( ρ + f ) decreases. Thus due to convergence cyclonioc vorticity increases. Solenoidal term:- As explained in the context of circulation theorem, here also solenoidal term signifies the contribution of the baroclinic effect of atmosphere towards the production or destruction of absolute vorticity. Let us consider the first term in the solenoidal term, the 1 ∂ρ ∂p term ⋅. ρ 2 ∂x ∂y 14 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. Now as per the equation there will be generation of cyclonic vorticity if ∂ρ ∂p >0 and > 0. Now question is what is the physical mechanism for that. ∂x ∂y Consider the adjoining fig1.11. In this figure a rectangular horizontal plane has been considered, which has been divided into two parts, Eastern part having more density (ρ) than the western part. ∂ρ In conformity with the condition > 0. We also consider that ∂x ∂p pressure is increasing towards north (Q > 0 ). Hence Pressure gradient force ∂y 1 ∂P is directed from North south. Since PGF= − , hence the western part of the ρ ∂y plane will be exerted by a higher PGF than the eastern part. This difference in PGF creates a torque which makes the plane to rotate in an anticlockwise direction. as shown in this figure. Thus cyclonic vorticity is generated. Similarly the other term, can also be explained. Tilting term: This term explains the generation or destruction of the vertical component of vorticity by the tilting of horizontal vorticity due to non uniform vertical motion. ⎡ ∂w ∂v ∂w ∂u ⎤ Tilting term:- −⎢ ⋅ − ⋅ ⎥ ⎣ ∂x ∂z ∂y ∂z ⎦ ∂w ∂v We consider the first term, − ⋅ ∂x ∂z ∂v ∂w If 0. ∴ DH > 0 , implying that at 300 mb divergence takes place in this region. Hence low pressure area forms at the surface area ahead of an upper air trough. Similarly in (C) region winds coming from a ridge, a source of anticyclonic vorticity, hence anticyclonic vorticity advection takes place over this region. Hence in this region r − V H ⋅ ∇ H (ζ + f ) < 0 ; so f ( DH ) = −VH ⋅ ∇ H (ζ + f ) < 0 ∴ DH < 0 , so there is convergence behind the trough at 300 hPa, so high pressure area forms at the surface behind an U.A. trough. We shall discuss the divergence pattern in different sectors of Jet Stream. To discuss the divergence pattern in different sectors of the sub-tropical westerly jet stream, we may refer figure 1.15. In this figure four sectors have been shown Sector I (Left exit) In this sector we have considered two points P & Q, P being nearer the core and Q being away from Jet core. We compute the vorticity at these two points using natural co- ordinate. In the natural co-ordinate system, vorticity ζ is given by ∂V ζ = VK s − ; K s being the Stream line Curvature, here K s = 0 as the ∂n stream lines are almost straight line for Jet stream. ∂V ∴ζ = − ∂n Now in this sector, at the point P = + 27.5 Unit and Q = 22.5 unit, But the direction of wind is from P to Q ie. wind is coming from higher cyclonic vorticity to lower cycloniv vorticity. Hence in this case advection is cyclonic vorticity. ∴ −VH ⋅ ∇ H (ζ + f ) > 0 19 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. ∴ f ( DH ) = − V H ⋅ ∇ H (ζ + f ) > 0. So, DH > 0 Hence divergence takes place at the Jet Core Level in the Left exit sector I. Following a similar approach, divergence pattern in other sectors may also be found. Barotropic or Rossby potential vorticity: We consider a fluid flow in an infinite channel, bounded below by the earths surface and above by a rigid lid (for example, tropopause). For such fluid flow, normal component of fluid at any point is zero, i.e., at any point, Vn = 0. Hence, ( ) r r from Gauss’s divergence theorem we have, ∫∫∫ ∇.V dσ = ∫∫ Vn ds = 0. Hence such σ s flow is non divergent. Hence, for such flow the scaled vorticity equation reduces to: d (ς + f ) ∂w = (ς + f ). Since, the flow is non-divergent, we may ignore the effect dt ∂z of ageostrophic part of horizontal wind. Also we consider a barotropic atmosphere. Under these conditions, vertical integration of the above equation from z = z b to z = z t leads to h d (ς + f ) d z t d z b dh = w( z t ) − w( z b ) = − = , where, h = z t − z b is the depth (ς + f ) dt dt dt dt of the fluid. The above equation after integration with time further simplified to ς+f = Constant. This quantity is known as Barotropic or Rossby potential h vorticity. This is known as conservation of Barotropic potential vorticity. For non-divergent flow at any level, scaled vorticity equation reduces to d (ς + f ) = 0 , i.e., ς + f = constant. Trajectory of an air parcel conserving dt absolute vorticity is known as Constant Absolute Vorticity (CAV) trajectory. It can be shown that this trajectory is looked wave like. Baroclinic or Ertel’s potential vorticity To obtain an expression for Baroclinic or Ertel’s potential vorticity, we start from horizontal equation of motion in ( x, y,θ , t ) co-ordinate. We know that vector form of the horizontal equation of motion in isobaric co- ordinate is given by r r ∂VH ( ) ∂VH r r r r r r + V H.∇ P VH + ω = −∇ Pφ + fkˆ × VH + FH ∂t ∂p 20 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for Advanced (Revised) Meteorological training course-Phase II. r r r r ∂VH ( ) ∂V H ∂VH ( ) ∂ r r r r r r V It can be shown that, + VH.∇ P VH + ω = + VH.∇ θ VH + θ& H and ∂t ∂p ∂t ∂θ r r also it can be shown that, ∇ Pφ = ∇ θ M , where M = C P T + gz is Montgomery stream function. Hence, the vector form of horizontal equation of motion in isentropic co-ordinate is given by: r r ∂VH ∂t ( r r r ) ∂V r r r + VH.∇ θ VH + θ& H = − ∇ θ M + fkˆ × VH + FH …..(C1.27) ∂θ r Performing k̂ × ∇ θ on both sides of the above equation for frictionless flow we d (ς θ + f ) have, = − Dθ (ς θ + f ) …..(C1.28), dt where, Dθ , ς θ are respectively the horizontal divergence and vertical component of vorticity in isentropic co-ordinate. Again continuity equation in isentropic co- 1 dσ ∂p ordinate gives, = − Dθ …..(C1.29), where, σ = g −1. σ dt ∂θ Combinining (C1.28), (C1.29) and then integrating with respect to time we obtain ςθ + f = Constant. This is known as conservation of baroclinic potential vorticity σ and the quantity on the LHS is known as baroclinic potential vorticity. 21 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. Chapter-II PERTURBATION THEORY Main goal in Meteorology is to forecast the weather parameters for the future time with the knowledge of their present value. Bjerkness (1904) had recognized this problem of weather forecasting as an initial value problem(IVP). Initial value problem is a partial differential equation (Linear/ Non-Linear) with time (t) as an in dependent variable. Some Useful Concepts : Partial derivative: Let a quantity ‘S’ is dependent on x, y, z, t. Then derivative of S with respect to any one (say t) of these four, keeping rest three unchanged, is called partial derivative of S with respect to ‘t’. For example 24 hrs change of pressure at a place is the partial ∂s ∂s ∂s ∂s change in pressure with respect to time. These are denoted by , , , etc. ∂t ∂x ∂y ∂z Examples: Let, V = x 3 + y 3 + 3axyz ∂V Hence, = 3x 2 + 3ayz (y, z have been kept constant) ∂x ∂V = 3 y 2 + 3axz (z, x have been kept constant) ∂y ∂V = 3axy (x, y have been kept constant) ∂z Partial differential equation (PDE): A differential equation is an equation which involves derivative or differential of the dependent variable. A PDE is an equation which involves partial derivatives or differentials of the dependent variable. ∂u ∂u 1 ∂p EX: u +v =− + fv is a partial differential equation, as it contains the ∂x ∂y ρ ∂x partial derivatives of the dependent variables u, p. Order of a PDE : It is the highest order partial derivatives involved in the equation. 1 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. ∂ 2u ∂ 2u Ex. Consider the PDE + = F ( x, y ). ∂x 2 ∂y 2 Here, u is the dependent variable, x, y are independent variables and F(x ,y) is a known function of x,y. In the PDE the highest order partial derivative involved in this equation is 2. So the order of this PDE is 2. Linear and non-linear PDE : A general form of a 2nd order PDE is given by ∂ 2u ∂ 2u ∂ 2u ∂u ∂u A + B + C +D +E + Fu = G. ∂x 2 ∂x∂y ∂y 2 ∂x ∂y In the above equation A,B,C,D,E,F and G are called coefficients of the PDE. If all these coefficients are constants or functions of independent variables ( x , y), then the resulting PDE is known as a Linear PDE. For example let us consider the following PDE: ∂ 2u ∂ 2u ∂ 2u + 2 + =0. ∂x 2 ∂x∂y ∂y 2 For this PDE A=1 B=2 C = 1 and D = E = F = G=0. Hence this PDE is a Linear PDE. We consider another PDE, ∂ 2u ∂ 2u 2 ∂ u 2 y2 + 2 xy + x = ( x + y ). In this PDE, A, B, C and G are functions ∂x 2 ∂x∂y ∂y 2 of x or y or both. So, this is also a 2nd order linear PDE. on the other hand if at least one these coefficients is a function dependent variable, then the resulting PDE is known as a non-linear PDE. For example let us consider the following PDE: ∂u ∂u 1 ∂p u +v =−. ∂x ∂y ρ ∂x In the above equation, A = B = C = F = 0, D = u and E = v. Since u, v are dependent variables, hence it is a non-linear PDE. Need for the perturbation theory : 2 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. There are several method for weather forecasting, viz. synoptic, statistical, Dynamic (Numerical weather prediction) method etc. In the NWP, the governing equations are solved for the weather parameter, viz. u,v,w, T, P etc.. The governing equations are non-linear partial differential equation. Non-linear partial differential equations can not be solved exactly, as till now we don’t have any method to get exact solution of non-linear partial differential equation. To get rid of the above problem, there are two ways viz., (a) Transform the non-linear partial differential equation into ordinary differential equation and then get exact solution. (b) Transform the set of partial differential equations into their finite difference form and then solve them numerically. Discussion about (a) is beyond the scope of discussion. Now while discussing (b), it is worth mentioning that the numerical solution of these non-linear partial differential equation is highly sensitive to the initial conditions given, i.e. a slight change in the initial condition may lead to an abrupt change in the numerical solution. This is due to the presence of non-linearity in the governing equations. Perturbation theory was proposed to remove the non-linearity from the governing equations. Basic postulates of perturbation theory : This theory is based on same postulates, which are given below : I. According to this theory, the total atmospheric flow consists of a mean flow and a perturbation superimposed on it. So, that all field variables consist of a basic (mean) part and a perturbation part. II. Both the mean part and the total (mean + perturbation) satisfy the governing equations. Mean part is the temporal and longitudinal average of the variable as a result of which it is independent of x and t. III. The magnitude of perturbation part is very small as compared to that of mean part, so that any product of perturbations or product of their derivatives or product of a perturbation and derivative of perturbation may be neglected. Now, it is our task, to verify whether using the above postulates, the non-linearity from a term of governing equation may be removed or not. 3 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. ∂ϕ For that we consider an arbitrary non-linear term, say, u. ∂x Using postulate (I), u = u + u ′ and ϕ = ϕ + ϕ ′. ∂ϕ ∂ (ϕ + ϕ ′) ⎛ ∂ϕ ∂ϕ ′ ⎞ ∂ϕ ′ ∂ϕ ′ Hence, u = (u + u ′) = (u + u ′)⎜⎜ + ⎟=u ⎟ + u′ (Here, ∂x ∂x ⎝ ∂x ∂x ⎠ ∂x ∂x ∂ϕ ∂ϕ ′ = 0 , as per 2nd part of postulate (II)). Again using postulate (III), u ′ ≈ 0 , being a ∂x ∂x product of perturbation quantity and its derivative. Hence using perturbation technique, non-linearity from the governing equations may be removed. 4 Lecture notes on Dynamic Meteorology for Revised Advanced training course prepared by Dr.Somenath Dutta. Mechanisms of pressure change Pressure tendency equation: To derive pressure tendency equation, we shall start from the hydrostatic approximation ∂p = − g ρ …….(1) ∂z Integrating the above equation vertically from an arbitrary level z = z 0 to z = ∞ , we obtain, ∞ ∞ ∂p ∫ ∂z z = z0 dz = − ∫ gρ dz z = z0 ∞ ⇒ p( z 0 ) = ∫ g ρ dz , since, at the top of the atmosphere there is no pressure. z0 Now, differentiating the both sides of the above partially with respect to time, we obtain, ∂p ∂ ⎛⎜ ⎞ ∞ ∞ ⎟ = g ∂ρ dz ∂t ∂t ⎜⎝ z∫0 ∫ ∂t = g ρdz ⎟ ⎠ z0 ∂ρ r r Again from continuity equation we have, = −∇.( ρV ) ∂t ∞ r r ∂p = − g ∫ ∇.( ρV ) dz ∂t z0 So, we have, ∞ ∞ ( ) r r r r = − g ∫ ρ (∇ h. Vh )dz + g ∫ − Vh. ∇ h ρ dz + gρ ( z 0 ) w( z 0 ) z0 z0 The above equation is known as pressure tendency equation. Left hand side of the above equation represents pressure tendency at a point at level z = z 0 and right hand side consists of three terms each of which representing some mechanisms for pressure change. First term is known as divergence term. It represents net lateral divergence or convergence across the sidewall of an atmospheric column with base at z = z 0 and extending up to top of the atmosphere. We know that pressure at z = z 0 is nothing but the weight of air contained in an atmospheric column with base at z = z 0 having unit cross sectional area and extending up to top of the atmosphere. Now this weight will increase or decrease if mass of air inside this column increases or decreases. Again mass of air inside this column increases or decreases if there is net inflow (convergence) or out flow (divergence) of air. Hence, net lateral divergence leads to fall in pressure and net lateral convergence leads to a rise in pressure. For synoptic scale system, this term contributes significantly towards pressure change. Second term expresses the net lateral advection of mass into the atmospheric column with base at z = z 0 having unit cross sectional area and extending up to top of the atmosphere. Clearly net positive advection leads to an increase in mass, which in tern leads to rise in pressure and net negative advection leads to a decrease in mass which in tern leads to fall in pressure. Third term expresses flux of mass into the above atmospheric column across its base at z = z 0. Lecture notes on Dynamic Meteorology for Revised Advanced training course prepared by Dr.Somenath Dutta. Movement of different pressure systems: Here we shall discuss the movement of pressure systems (lows/highs) for different isobaric patterns. Mainly we shall discuss Sinusoidal pattern, circular pattern and circular pattern beneath a Sinusoidal pattern above. Sinusoidal isobaric pattern: Let us refer to the adjoining sinusoidal pressure pattern. Ahead of the trough there is divergence and ahead of the ridge there is convergence at the surface. Hence fall in pressure takes place ahead of trough and rise in pressure ahead of ridge. Due to this, after some time lowest pressure will be found ahead of trough, as a result trough will be shifted towards east of its present location. Hence, the pressure system will move in a westerly direction. 996 998 Fig.1 Circular low-pressure pattern: Let us consider the adjoining circular low- pressure pattern. Lowest pressure is at the center of the circular pattern. To the north of the center Coriolis force is higher than that to the south. As we know that Coriolis force makes flow anticyclonic, hence cyclonic wind will be more to the south than to the north of the center. Hence to the east of the center there is downstream decrease in wind speed and to the west there is down stream increase in wind speed. Hence divergence takes place to the west of the center as a result of which there will be fall in pressure to the west of the center. Due to this, center of low after some time will be shifted to the west of its present position. Hence net result is movement of the pressure system in an easterly direction. Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. Chapter-III ATMOSPHERIC WAVES: Wave may be defined as a form of disturbance in a medium. When a disturbance is given to a part of an elastic medium, then that part gets displaced from its original position. But by the virtue of elasticity, a restoring force is developed in the displaced part, which helps it to return to its original position. This leads to an oscillatory motion, which is known as wave. Some useful concepts on waves: WAVE LENGTH: It is defined as the distance between two consecutive points on the wave, which are in the same phase of oscillation, i.e. distance between two successive troughs or ridges. WAVE NUMBER: Wave number of a wave with wave length ‘L’ is defined as the number of such waves exist around a circle of unit radius. Hence wave number 2π k is defined by, k = , where L is the wave length. L Since a wave may travel in any direction, hence we may define wave length / wave number for three directions, viz. along x, y and z directions. If L x , L y and L z are respectively the wave lengths along x, y and z directions and if k, l and m are wave numbers along x, y and z directions, then 2π 2π 2π k= ,l = and m =. Lx Ly Lz FREQUENCY : It is the number of wave produced in one second.It is denoted by ν. PHASE VELOCITY: We know that any disturbance behaves like a carrier. So, wave may be thought of as a carrier. Phase velocity is defined as the rate at which Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. momentum is being carried by the wave. For practical purpose, it may be taken as the speed with which a trough / ridge moves. It can be shown that, phase velocity in any direction = frequency / wave r number in that direction. Thus if the phase velocity vector C has components Cx, Cy, Cz along x, y and z direction, then C x = ν / k , C y = ν / l and C z = ν / m. GROUP VELOCITY : It is the rate at which energy is being carried by the wave. When a single wave travels then the energy and momentum are carried by the wave at the same rate. But when a group of wave travel then momentum propagation rate and energy propagation rates are different. So, in such case group velocity and phase velocity are different. Thus r if the phase velocity vector C G has components CGX, CGY, CGZ along x, y and z ∂ν ∂ν ∂ν direction, then C GX = , C GY = and C CZ =. ∂k ∂l ∂m DISPERSION RELATION : It is a mathematical relation ν = f (k , l , m) between the frequency ( ν ) and wave numbers k , l , m. Generally for any wave, phase velocity and group velocity is obtained from the dispersion relation. If for any wave phase velocity and group velocity are same, then it is called a non-dispersive wave, otherwise it is a dispersive wave. ROSSBY WAVE: First it will be shown how conservation of absolute vorticity (ζ+f) leads to wave like motion. We consider an object placed on or over the earths surface at latitude ‘ϕ’. In the adjoining figure, a meridional circle passing through the Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. object has been shown. Suppose, while motion, the absolute vorticity (ζ+f) of the object remains conserved. Let the object be at stationary state initially. Then the relative vorticity ( ζ ) of the object is zero at the initial state. Let f 1 be the value of planetary vorticity at the initial state. Now if the object be displaced meridionally, then its relative vorticity will change to ζf (say). If f2 be the value of planetary vorticity (f) at the final state, then we must have ∂f 0 + f 1 = ς f + f 2 ⇒ ς f = −( f 2 − f1 ) = −δf = − δ y = − βδ y. ∂y Hence, ς f > 0 , if δy < 0, i.e., for a southward displacement and ς f < 0 , if δy > 0, i.e., for a northward displacement. So, if the object is displaced northward, then it turns anti- cyclonically towards its initial latitude. At the initial latitude ζf = O, but by inertia it will continue to move southward, cross the initial latitude and acquire cyclonic vorticity. After acquiring cyclonic vorticity, the object turns towards its original latitude. Thus the object executes wave like motion about its initial latitude ‘ϕ’. This wave is known as Rossby wave. Thus the dynamical constraint for Rossby wave is the conservation of absolute vorticity. So, to obtain the dispersion relation for the Rossby wave, the governing equation is conservation of absolute vorticity , i.e. d (ς + f ) ∂ς r r =0⇒ + V.∇ς + vβ = 0 …..(1) dt ∂t The above equation is linearised using perturbation method. Here we made the following assumptions : „ Atmosphere is auto-barotropic „ Basic flow is zonal „ Basic zonal flow is meridionally uniform Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. With these assumptions, the above governing equation may be linearised to ∂ς ′ ∂ς ′ +U + v ′β = 0 ….(2) ∂t ∂x ς ′ is perturbation relative vorticity and v ′ v’ is perturbation meridional wind. For equation (2) we seek for wave like solution, like, ( )′ ∝ e i ( k x −c t ) , where, k is the zonal wave number and c is the zonal phase velocity. β After simplification we obtain following dispersion relation , ν = U k −. k ν β ∂ν β Hence phase velocity C = =U − and group velocity CG = =U + 2. k k 2 ∂k k Clearly C ≠ C G. So, Rossby wave is a dispersive wave. β Since C − U = − , hence Rossby wave retrogates with respect westerly k2 β mean flow. Again C G − U = > 0. Hence Rossby wave carries energy in the k2 downwind direction with respect to westerly mean flow. Physically the above results may be interpreted as follows: For momentum source is the westerly mean flow and for energy the source is the disturbance i.e., the wave. HAURWITZ WAVE : This wave is a generalization of the Rossby wave. Similar to Rossby wave, this wave also results from the conservation of absolute vorticity. To obtain the dispersion relation for this wave we take the same assumptions as in Rossby wave except that, here we assume that the basic zonal flow ‘U’ is not uniform in the meridional direction, rather it is a function of ‘y’ (latitude) and amplitude of this wave is zero at y = ± d, i.e., U(± d) = 0. Starting with the conservation of absolute vorticity, and following the approach, similar to that, made in Rossby wave, we arrive at the following dispersion relationship. Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. ⎛ d 2U ⎞ ⎜⎜ β − 2 ⎟⎟ k ν =U k − ⎝ dz ⎠ ⎛ 2 π2 ⎞ ⎜⎜ k + 2 ⎟⎟ ⎝ 4d ⎠ Clearly if the zonal basic flow ‘U’ is uniform in the meridional d 2U β direction, then 2 = 0 and d → ∞. In that case ν = U k −.This is nothing but dz k the dispersion relationship for Rossby wave. So, the Haurwitz wave is a generalisation of Rossby wave. GRAVITY WAVE We have seen that to generate any wave always a restoring force is required. Gravity waves are those waves, for which he restoring force is buoyancy. Classification of Gravity waves: Gravity waves External Gravity Wave Internal Gravity Wave Can travel along the interface Can travel along the interface between between two fluids of different two fluids of different densities. densities. Cannot travel across the fluid. Can also travel across the fluid. Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. Vertical scale is negligible Vertical scale is comparable to the Compared to horizontal scale horizontal scale of motion of motion Eg. Sea waves, Tsunami Eg. Mountain wave, etc. EXTERNAL GRAVITY WAVE (EGW) To study the external gravity wave, we consider two different fluids of densities ρ1 and ρ 2 ( ρ1 > ρ 2 ) placed one over the other. In the undisturbed condition their interface is a plane surface whose vertical section is a horizontal line as shown in figure 3(a). Now if any perturbation is given to the interface, then it would no longer be a plane surface, rather a wavy surface. Its vertical section would be a wave as shown in fig. 3(b). To study this wave, we consider wave motion in the x-z (Zonal-vertical) plane, as shown in fig. 3(c). The governing equations are: u-momentum equation, continuity equation. These equations are linearized using perturbation method. Then wave like solution is sought for the perturbation height of the interface. Then after simplification we obtain the following dispersion relation. ∆ρ ν = Uk ± k gH , where, H is the mean depth of the free surface, ρ1 ∆ρ = ρ1 - ρ 2. Now if we take air over ocean water, then definitely ρ1 >> ρ 2 and ∆ρ = ρ1 - ρ 2 ≈ ρ1 , and in that case ν = Uk ± k gH ν Hence, phase speed C = = U ± gH k Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. ∂ν And group velocity C G = = U ± gH. ∂k Hence C = CG So, EGW is a non-dispersive wave. Here gH is known as shallow water gravity wave speed and U is known as Doppler shift. Internal gravity wave (IGW) : To study IGW we consider, for simplicity, a flow which is, 2 – D (x-z) Adiabatic Frictionless Non – rotational Boussinesq. The governing equations are: U-momentum equation W-momentum equation Continuity equation Energy equation under adiabatic condition. The above equations are linearised using perturbation method. The linearised form of the above equations are : ∂u ′ ∂u ′ 1 ∂p ′ +U =− ∂t ∂x ρ 0 ∂x ∂w′ ∂w′ 1 ∂p ′ θ′ +U =− +g ∂t ∂x ρ 0 ∂z θ0 Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. ∂u ′ ∂w′ + =0 ∂x ∂z ∂θ ′ ∂θ ′ +U =0 ∂t ∂x Wave solutions, for the perturbations in the above equations are sought. Wave solutions are like exp [i (kx + mz − ν t )] Then after same simplifications we obtain the following dispersion relationship Nk ν = Uk ±. k 2 + m2 Phase velocity : ν N X-Component of phase velocity C x = =U ± , k k 2 + m2 ν Uk Nk Z-Component of phase velocity C z = = ±. m m m k 2 + m2 Group velocity: ∂ν Nm 2 X-Component of group velocity C Gx = =U ± , ∂k k 2 + m2 ∂ν ⎛ Nkm ⎞ Z-Component of group velocity C Gz = = −⎜⎜ ± ⎟. ⎟ ∂m ⎝ k 2 + m2 ⎠ Now we consider a special case for U = 0 Nk ⎛ Nkm ⎞ Then C z = ± and C Gz = −⎜⎜ ± ⎟ ⎟ m k 2 + m2 ⎝ k 2 + m2 ⎠ Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. Thus it follows that for a given combination of signs of k , l , ; C Z and C GZ are opposite to each other. Thus vertical phase propagation (momentum propagation) and group propagation (energy propagation) by IGW are opposite to each other. Also from the above expressions of C’s and CG ‘s it follows that the vector r ) C = i C X + ˆjCY is perpendicular to the phase lines kx + mz − ν t = constant, where r ) as the vector C G = i C GX + ˆjC GY is parallel to the phase lines kx + mz − ν t = constant. Hence for the IGW, phase velocity and group velocity are perpendicular to each other. Importance of IGW: IGW, although, is generated at lower troposphere, they can transport energy, momentum etc upto a great height. From the expressions for phase velocity and group velocity, it is seen that a vertically propagating IGW extracts Westerly Momentum from the mean flow at upper level or imparts easterly momentum to the mean flow at upper level. IGW is believed to be one of the causes responsible for QBO. CAT is believed to be also due to continuous extraction of momentum from upper level Lecture notes on Dynamic Meteorology prepared by Somenath Dutta for advanced (old) meteorological training course. mean flow by  Chapter-IV Planetary Boundary Layer (PBL) A Brief essay on PBL: PBL is the lower most portion of the atmosphere, adjacent to the earth’s surface, where maximum interaction between the Earth surface and the atmosphere takes place and thereby maximum exchange of Physical properties like momentum, heat, moisture etc., are taking place. Exchange of physical properties in the PBL is done by turbulent motion, which is a characteristic feature of PBL. Turbulent motion may be convectively generated or it may be mechanically generated. If the lapse rate near the surface is super adiabatic, then PBL becomes positively Buoyant, which is favourable for convective motion. In such case PBL is characterized by convective turbulence. Generally over tropical oceanic region with high sea surface temperature this convective turbulence occurs. If the lapse rate near the surface is sub adiabatic then the PBL is negatively buoyant and it is not favourable for convective turbulence. But in such case, if there is vertical shear of horizontal wind, then Vortex (cyclonic or anti cyclonic) sets in, in the vertical planes in PBL, as shown in the adjacent fig 2b. This vortex motion causes turbulence in the PBL, known as mechanical turbulence. If the PBL is positively buoyant as well as, if vertical shear of the horizontal wind exists, then both convective and Mechanical turbulence exits in the PBL. The depth of the PBL is determined by the maximum vertical extent to which the turbulent motion exists in PBL. On average it varies from few cms to few kms. In case of thunderstorms PBL may extend up to tropopause. Generally at a place on a day depth of PBL is maximum at noon and in a season it is maximum during summer. Division of the PBL into different sub layers: The PBL may be sub divided into three different sections, viz viscous sub layer, the surface layer and the Ekmann layer or entrainment layer or the transition layer. Viscous layer is defined as the layer near the ground, where the transfer of physical quantities by molecular motions becomes important. In this layer frictional force is comparable with PGF. The surface layer extends from z = z 0 (roughness length) to z = z s with z s , the top of the surface layer, usually varying from 10 m to 100 m. In this layer sub grid scale fluxes of momentum (eddy stress) and frictional forces are comparable with PGF. The last layer is the Ekmann layer is defined to occur from z s to z i , which ranges from 100 m or so to several kilometers or more. Above the surface layer, the mean wind changes direction with height and approaches to free stream velocity at the height z as the sub grid scale fluxes decrease in magnitude. In this layer both the COF and Eddy stress are comparable with PGF. Boussinesq approximation: According to this approximation density may be treated as constant everywhere in the governing equations except in the vertical momentum equation, where it is coupled with Buoyancy term. Physically this approximation says that the variation of density in the horizontal direction is insignificant as compared to that in the vertical direction. Governing equations in the PBL: Governing equations in the PBL, following adiabatic and Boussinesq approximation, are given below: ∂u r r ∂t ( ) + V.∇ u = − 1 ∂p ρ 0 ∂x + fv + Fx ……..(4.1) ∂v r r ∂t ( ) + V.∇ v = − 1 ∂p ρ 0 ∂y − fu + F y ……(4.2) ∂w r r ∂t ( ) + V.∇ w = − 1 ∂p ρ 0 ∂z −g θ θ0 + Fz ……(4.3) ∂u ∂v ∂w + + = 0 …….(4.4) ∂x ∂y ∂z ∂θ ( ) rr + V.∇ θ = 0. ……(4.5) ∂t Concepts of mean motion and eddy motion in the PBL & Reynolds averaging technique. In the PBL both the mean motion and the eddy motion are very important. Hence it is required to have equations for both motion. To distinguish these two, Reynold devised an averaging method, which is discussed below: Let us consider any field ‘ S ’at a synoptic hour T. Let S Obs bet he observed value of ‘ S ’ at time T hrs. Now to find out the contribution from mean and eddy motion towards ‘ S ’, we have to ⎛ τ τ⎞ take a number of observations of ‘ S ’ during the time interval ⎜ T − , T + ⎟. Suppose during the above ⎝ 2 2⎠ τ T− 2 1 n period we have ‘n’ observations. Viz., S 1 , S 2 ,....., S n of S. Then S = ∫ s dt ≈ ∑ S i is called the n 1 τ T− 2 ( ) mean part of ‘S’ at T, and S ′ = S obs − S is called the eddy part of S at time T hrs. This Eddy part is 2 due to turbulent eddy motion in the PBL. The quantity ‘τ’ is called averaging interval. While choosing ‘τ’ the following precautions are necessary to take: a) It should not be too small to miss the trend in mean motion. b) It should not be too large that eddies filtered out. For two arbitrary quantity, say, α and β, we have, α = α +α ′ and β = β + β ′. Hence, αβ = α β + α ′β ′. The last term is known as eddy co-variance. Concept of Eddy flux and Eddy flux divergence/ convergence: Flux of any field refers to the transport of that field in unit time across unit area. Hence flux of a r r field, say S , is SV , V being wind velocity. Eddy flux, thus refers to the transport of some field by eddy wind. If u ′, v ′, w ′ are the r components of eddy wind, then eddy wind vector is given by V ′ = (iˆu ′ + ˆjv ′ + kˆw′) , then eddy flux of a r quantity S is SV ′. Flux divergence/convergence physically refers to the dispersion or accumulation of the field after r r being transported. Mathematically it is expressed as ∇.( SV ′). In the mean equations of motion some new terms have appeared. These terms are known as eddy flux convergence of eddy momentum. Physically they may interpreted as follows: Let us consider, the eddy zonal momentum (u ′) is being transported by all the three components u ′, v ′, w′ of eddy wind. Now eddy zonal momentum transported by these components in unit time across unit area are respectively u ′u ′, u ′v ′ and u ′w′. The first one is along iˆ direction, second one in ĵ direction and third one in k̂ direction. Thus at ( ) r any point transport of u ′ may be expressed as the vector u ′V ′. After being transported, the eddy u momentum is being accumulated, which is expressed as ( ) r r − ∇. u ′V ′. This term is called eddy flux convergence of u ′. Thus, this much eddy zonal momentum is being added to the existing mean zonal momentum u, causing a change in u. Thus this term has appeared in 3 the zonal momentum equation for the mean flow. Similarly one can argue for the existence of the other eddy flux convergence terms. Governing equations for mean motion: To obtain the equations for mean flow, we first need to express rr (V.∇)u in flux form. terms like, ( ) rr r r r r r r We know that, (V.∇)u = ∇. uV − u∇.V. Again following Boussinesq approximation, ∇.V = 0. ( ) rr r r r ( ) [ r r ( )] ( ) r r r r ( ) r r r ( ) r Hence, V.∇ u = ∇. uV = ∇. (u + u ′) V + V ′ = ∇. u V + ∇. u V ′ + ∇. u ′V + ∇. u ′V ′ ( ) Taking Reynolds average, we have, (V.∇)u = ∇.(u V )+ ∇.(u ′V ′) rr r r r r Again ∂u ∂u ∂u ′ ∂u = + ∂t ∂t ∂t = ∂t and, ∂u r r ∂u ∂t + ∇. u V = ∂t ( ) r r ( ) ( ) r r ∂u + V.∇ u + u ∇.V = ∂t r r + V.∇ u ( ) Hence, the governing equations for mean flow are ∂u ∂t ( ) r r + V.∇ u = − 1 ρ0 ∂p ∂x r ( ) r + fv + Fx − ∇. u ′V ′ ……. (4.6) ∂v r r ∂t ( ) + V.∇ v = − 1 ρ0 ∂p ∂y r ( )r − fu + F y − ∇. v ′V ′ ……..(4.7) ∂w r r ∂t ( ) + V.∇ w = − 1 ∂p ρ 0 ∂z −g θ θ0 r r ( ) + Fz − ∇. w′V ′ ……(4.8) ∂θ ∂t ( ) r r r ( ) r + V.∇ θ = −∇. θ ′V ′ …..(4.9) r r ∇V. = 0 …..(4.10) Turbulent Kinetic Energy Equation Turbulent Kinetic Energy equation is obtained from the equations of motion , in component form, for turbulent motion, which can be obtained by subtracting equations 4.6, 4.7 & 4.8 from 4.1, 4.2 & 4.3 respectively. Then the subtracted equations are multiplied by u ′ , v ′ , w′ respectively, then they are added and then taking Reynolds average we obtain required TKE equation ∂ (TKE) = MP + BPL + TR − ε ……(4.11) ∂t 4  r Where, BPL = g θ0 r w′ θ ′ = − N 2 ξ ′ w′ , and MP = − w′ V ′ ∂V ∂z ( ). Here, BPL stands for Buoyancy production or loss, MP stands for Mechanical production, N 2 stands for square of Brunt-Vaisalla frequency and ξ′ is eddy vertical displacement. TR stands for redistribution by transport and pressure forces and ε represents frictional dissipation which is always positive reflecting the dissipation of the smallest scale of turbulence by molecular viscosity. In our introduction we have mentioned that generally turbulence in PBL is either convectively or mechanically generated. Now we shall see that, for convectively generated turbulence, through BPL term eddies are being supplied K.E. Effect of Buoyancy production or loss (BPL) term : To examine, the effect of this term, we shall consider three conditions viz., When atmosphere is stably stratified, When atmosphere is unstably stratified and When atmosphere is neutrally stratified. First of all we must note that eddy co-variance between eddy vertical velocity and vertical displacement must be positive, as the former one must be upward or downward if the later is so. If the Atmosphere is stably stratified, then we know that N 2 is positive. Hence in that case BPL must be negative. Thus convective turbulence is suppressed in a stably stratified PBL. Similarly one can show that 2 in an unstably stratified PBL ( N < 0), BPL is positive and convective turbulence is sustained. Finally if the PBL is neutrally stratified, then N 2 = 0. So BPL = 0, hence Convective turbulence is neither generated nor sustained. Effect of Mechanical Production (MP) term: In the introduction it was shown qualitatively that Mechanically generated turbulence can occur only if there is a vertical shear (either cyclonic or anticyclonic) of the horizontal wind. Now we can discuss the MP term and see how it is significant for mechanically generated turbulence. First term of MP represents the vertical eddy flux of eddy horizontal momentum and the second one is vertical shear of eddy horizontal momentum (i.e., vertical gradient of the components of mean horizontal wind). Qualitatively one can argue that if the vertical gradient of any quantity is positive (i.e., upward), then eddy transport of that quantity has to be downward and the vice-versa. Thus we see that vertical gradient of the mean and vertical eddy transports are opposite to each other. As a result of which MP is always positive, provided there is a vertical shear of mean horizontal wind. Hence in any case, due to MP term TKE increases with time and Turbulence is sustained. Thus, whenever there is vertical shear of the mean horizontal wind, then mechanically generated turbulence occurred. 5  Now, we consider a typical situation, when PBL is stably stratified is and there exists vertical shear of mean horizontal wind. In such situation BPL term inhibits turbulence and MP term enhances turbulence. In this situation it is difficult to say whether their combined effect is to suppress turbulence or to sustain turbulence. It has been found empirically that to maintain the turbulence, MP should exceed four times the BPL. This condition is measured by a quantity called the flux Richardson number (Rf), which is BPL defined by R f =−. If the PBL is unstably stratified then R f < 0 and turbulence is sustained by MP convection, as mentioned earlier. If PBL is stably stratified, then R f > 0. In such case R f must be less than 0.25 to sustain the turbulence. Thus R f should be less than ¼ to maintain turbulence in a stably stratified PBL by wind shear. 6 Sub-grid scale Physical processes: A Physical process whose spatial dimension is less than the grid scale, is known as sub-grid scale Physical process. Sub grid scale physical processes may be taken place in a smaller area, but its impact may be significant on the large scale flow. Tto illustrate it we give the following example:- Let a small region be conditionally unstable. Now, in this region moist convection is taking place i.e. moist air parcel is rising. Now at a level where all the moistures inside the air parcel has condensed releasing latent heat. That released Latent heat, which may be due to a sub grid scale physical process, viz, moist convection, will in turn heat up the atmosphere at that level, which may affect the temperature at the grid point. Thus, we see that though a physical process is capable of escaping from being caught at the grid points, it affects the field value at the grids. So if the effects of sub grid scale physical process cannot be incorporated in the NWP model, then forecast issued by NWP is definitely going to be wrong. Parameterization of sub grid scale physical processes in the PBL. To parameterize the sub grid scale physical processes in the PBL first we should know that what sub grid scale physical process is taking place in the PBL. The only sub grid scale physical process taking place in PBL is the vertical eddy transport (Order of magnitude of the horizontal eddy transport is very less compared to the vertical eddy transport.). Again the method of parameterization of eddy transport depends on the nature of PBL. The vertical profile of the horizontal components of the mean wind using the paramaterisation scheme. For the special case of horizontally homogeneous turbulence above the viscous sub-layer, molecular viscosity and horizontal turbulent flux divergence terms can be neglected. The mean horizontal equations of motion then become ∂u ∂t ( ) r r + V.∇ u = − 1 ∂p ρ 0 ∂x + fv − ∂ (u ′w′) ∂z ……(4.12) ∂v ∂t ( ) r r + V.∇ v = − 1 ∂p ρ 0 ∂y − fu − ∂ (v ′w′) ∂z ……(4.13) For the mid latitude synoptic scale system, we know that to a first approximation, acceleration term may be neglected as compared to the Coriolis force and Pressure gradient force terms. Out side the PBL, this approximation simply results into geostrophic balance. Inside PBL also the acceleration terms are still small compared to the Coriolis force and Pressure gradient force terms, but the turbulent flux term must be 7 retained. So, there is an approximate balance in the PBL between the pressure gradient force,Coriolis force and the eddy stress of the mean flow. ∂ (u ′w′) Thus we have, 0 = f (v − v g ) − ……(4.14) ∂z ∂ (v ′w′) 0 = − f (u − u g ) − ……(4.15) ∂z Parameterization of eddy stress in an unstably stratified PBL. As we know in such case PBL is dominated by convective turbulence. Due to convection, the PBL is well mixed i.e. the mean quantities remain almost invariant with height in the PBL. In such case to a first approximation it is possible to treat the layer as a slab, in which mean horizontal wind, potential temperature remains invariant in the vertical and turbulent fluxes vary linearly with height. For simplicity, one can assume that at the top of CBL, turbulent vanishes. Observations indicate that the surface momentum flux can be represented by a bulk aerodynamic formula (u ′w′) s = −C d (u 2 ) + v 2 u ….(4.16) and (v ′w′) s = −C d (u 2 ) + v 2 v ….(4.17). So integrating equations (4.14) and (4.15) in the vertical between the bottom [generally taken at Z =0 ] and top of the boundary layer, we obtain v= Cd fh (u 2 ) + v 2 u ….(4.18) Cd And u =ug − …. (4.19), where h is the height of the boundary layer. fh Parameterization of eddy stress in a stably stratified PBL. In a stably stratified PBL, the mean quantities do vary in the vertical. In such case vertical eddy tranport of any quantity ‘S’ is parameterized using K-theory/similarity theory /flux-gradient theory. According to this theory, vertical eddy transport of any physical quantity is proportional to the vertical gradient of the mean of that quantity and directed down the gradient, i.e., ∂u u ′ w′ = − K m …. (4.19), ∂z 8  ∂v v′ w′ = − K m …. (4.20) ∂z ∂θ and θ ′ w′ = − K h …. (4.21), where, K m and K h are constants, known as eddy coefficients. In this ∂z theory they are treated to be invariant in the vertical, which is a limitation of this theory. Mixing length theory: This theory was proposed by Prandtl. This theory is for a stably stratified PBL. We consider an eddy in the PBL, initially embedded in the mean flow at the same level. If the eddy is displaced vertically, then it will carry the physical properties of the mean flow of the old level. After some eddy vertical displacement, say, ' l ′' , the eddy mixes with the mean flow at a new level, imparting all its physical properties to it. This causes a fluctuation in the physical properties at the new level. As per mixing length theory, fluctuation in the physical property is proportional to ' l ' and to the vertical gradient of the mean of the physical property, i.e. for an arbitrary physical property ‘S’ we have ∂S S ′ = −l ′. We know that in the PBL the order of magnitude of the vertical motion of mean flow is ∂z w ≈ V , where, V = (u + v ) 1 2 2 2 comparable with that of horizontal motion, so,. ∂S ∂V Hence, S ′w′ = − l ′ 2 …..(4.22). ∂z ∂z Again from K- Theory, we have, ∂S S ′w′ = − K ∂z ∂V Combing these two theories, we have, K = l ′ ….(4.23). The parameter L = l ′ , is known as 2 2 ∂z mean mixing length. It is analogous to mean free path in the kinetic theory of gas. It is a measure of the eddy size. Thus the value of K is large for large eddies and for large vertical shear of mean horizontal wind. Thus eddy transfer, is more for larger eddy greater vertical shear of horizontal wind. 9 Ekmann layer It is also known as entrainment layer. In this layer there is approximately a balance between the pressure gradient force, coriolis force and eddy stress. Using the geostropic approximation at the top of PBL and from equations (4.19) & (4.20) we have ∂ 2u 0 = f (v − v g ) + K m …..(4.24) ∂z 2 ∂ 2v and 0 = − f (u − u g ) + K m …..(4.25) ∂z 2 The above equations are solved using the following boundary conditions: (1). u ( z ) = v ( z ) = 0 at z = 0 (2). As z → ∞, u → u g and v → v g. Adding equation (4.24) with i ( = − 1 ) times the equation (4.25) results into ∂ 2C i f −i f − C= C g ….(4.26), where, C = u + iv and C g = u g + iv g. ∂z 2 Km Km The general solution of (4.25) consists of two parts, Viz., the complementary function (CF) and the particular integral (PI). CF = A exp[(1 + i )γ z ] + B exp[− (1 + i )γz ] …..(4.27), where, γ = f. 2K m 1 if PI = (− C g ) = C g …..(4.28). if Km D2 − Km Thus general solution is given by, C = A exp[(1 + i )γ z ] + B exp[− (1 + i )γz ] + C g ….(4.29). The arbitrary constants A and B are determined from boundary conditions (1) and (2). Accordingly, A = 0 and B = −C g. Hence, the particular solution is given by C = C g [1 − exp{− (1 + i )γz}] ….(4.30). Now separation of the real and imaginary part on both sides of equation (4.30) results into, [ ] u = u g 1 − e −γ z cos(γz ) − v g e −γ z sin (γz ) …..(4.31) v = u g e −γ z sin (γz ) + v [1 − e