Mathematical Concepts & Notation

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1 Mathematical Concepts & Notation Appendix A: Notation x, δx: a small change in x t : the partial derivative with respect to t holding the other variables fixed d : the time derivative of a quantity that is a function of time only ẋ: time derivative of x, t ẍ: second-order time derivative of x, 2 x t 2 f x or x f: derivative of f with respect to x D Dt : the Material (Lagrangian Derivative, the time rate of change of a characteristic following a fluid element Below, bold denotes vector fields, α is a constant, and all other variables are scalar fields. The components of a vector field f are denoted as (f x, f y, f z. In addition, we have used the convention in atmospheric science where the 3-dimensional wind vector v = (u, v, w and the horizontal wind vector u = (u, v, where u = dx is the wind in the local east-west direction, v = dy is the wind in the local north-south direction and w = dz is the wind in the local vertical. Finally, let î, ĵ, ˆk be local unit vectors in the direction of the coordinates x, y, z. Appendix B: Vector Calculus Vector Operators Useful quantities Vector Formulas Below, a, b, c are vector fields, ψ is a scalar field and α is a constant. a (b c = b (c a = c (a b a (b c = b(a c c(a b (B.1 (B.2 E. A. Barnes 1 updated 06:13 on Wednesday 7 th October, 2015

2 Name Vector/Scalar Vector Notation Cartesian Expansion Gradient vector î + ĵ + ˆk 2-D Laplacian (scalar scalar 2 = u Divergence of a vector field scalar v + v + w Curl of a 2-D vector field vector u Curl of a 3-D vector field vector v ( v u ( w v ˆk î + ( u w ĵ + ( v u ˆk 2-D Jacobian determinant scalar J(ψ, ζ ψ ζ ψ ζ Name Relationship Cartesian Coordinates ( vorticity ω = v ω = w v î + ( u w ( ĵ + v ˆk vorticity in ˆk direction ω z = ζ = u ζ = v u streamfunction ψ, 2 ψ = ζ, v = ˆk ψ u = ψ, v = ψ (a b (c d = (a c(b d (a d(b c (B.3 (αψ = α ψ (a + b = a + b (ψa = ψ a + a ψ (ψa = ψ a + ψ a ( a = 0 ( ψ = 0 (a b = (a b + b a (B.4 (B.5 (B.6 (B.7 (B.8 (B.9 (B.10 (a a = ( a a + 1 (a a (B.11 2 E. A. Barnes 2 updated 06:13 on Wednesday 7 th October, 2015

3 Appendix C: Material Derivative 2-D material derivative of a scalar field Dρ Dt = ρ t + dx ρ + dy ρ = ρ t + u ρ + v ρ = ρ t + u ρ (C.1 (C.2 The operators v ( and u ( are important to know - these are advection operators in 3-D and 2-D respectively. 3-D material derivative of a scalar field DΦ Dt = Φ t + dx Φ + dy Φ + dz = Φ t + v Φ Φ = Φ t + u Φ + v Φ + w Φ (C.3 (C.4 3-D material derivative of a vector field Df Dt and for a particular component of f = f t + dx f + dy f + dz f = f t + u f + v f + w f = f t + v f Df x Dt = fx t + u fx + v fx + w fx (C.5 (C.6 (C.7 Advective vs Flux Form The flux form of a quantity may be written as: t ρ + (ρv The advective form of a quantity may be written as: Dρ Dt + ρ v Using B.6 and C.2, it can be shown that C.8 and C.9 are identical. (C.8 (C.9 E. A. Barnes 3 updated 06:13 on Wednesday 7 th October, 2015

4 Appendix D: Theorems Divergence theorem (Gauss theorem The divergence theorem relates the integral of the divergence of a vector field a over a volume V to the total flux of the vector field out of the closed surface A which surrounds the volume V: a dv = a n da, (D.1 where a is an arbitrary vector field and n is the unit outward normal to the surface A. Stokes theorem Stokes theorem is the curl analogue of Gauss theorem. It relates the integral of the normal component of a over the open surface A to the line integral of a around the perimeter bounding the surface A: ( a nda = a dr. (D.2 Appendix E: Differential Operators in Curvilinear Coordinate Systems Generalized vertical coordinates Consider a general vertical coordinate r, which is assumed to be a monotonic function of height z. If you wish to transform the equations of motion into a coordinate system (x, y, r, t from (x, y, z, t, then z = z(x, y, r, t becomes a dependent variable. Any function A = A(x, y, z, t = A(x, y, z(x, y, r, t, t can be transformed into the new coordinate system, where the horizontal and time derivatives transform as A s = A r s + A z (E.1 s s where s denotes x, y or t and a explicitly denotes a differential with a held constant. The vertical partial derivative of A follows simply from the chain rule. That is, A r = A r r. The gradient of A along a surface of constant r is thus, r r A = z A + A r r rz. It is important to remember that the directions of the unit vectors (î, ĵ and ˆk are the same in both the r and z coordinate systems. That is, the x and y axes are always horizontal and are not oriented along surfaces of constant r and the vertical axis is still perpendicular to x and y. (E.2 (E.3 E. A. Barnes 4 updated 06:13 on Wednesday 7 th October, 2015

5 General curvilinear coordinates The metric expression in general curvilinear coordinates x 1, x 2, x 3 is (dl 2 = (h 1 dx (h 2 dx (h 3 dx 3 2, (E.4 where dl is an element of length and h 1, h 2, h 3 are the metric coefficients. Let i, j, k be local unit vectors in the direction of the coordinates x 1, x 2, x 3. The gradient of a scalar p, the divergence and curl of a vector v (with components u, v, w are then given by i h 2 h 3 p p p = i + j p, h 1 1 h 2 2 h 3 3 [ 1 (h2 h 3 u v = + (h 1h 3 v + (h 1h 2 w h 1 h 2 h ] + j [ (h1 u (h ] 3w h 3 h h 1 h 2 [ (h3 w 2 (h 2v 3 (E.5 ], (E.6 [ (h2 v (h ] 1u. (E Formula (E.4 can also be written in the determinant form h 1 i h 2 j h 3 k 1. (E.8 h 1 h 2 h h 1 u h 2 v h 3 w Cartesian coordinates Now let s apply these general formulas to cartesian coordinates. The metric expression in cartesian coordinates is (dl 2 = (dx 2 + (dy 2 + (dz 2, (E.9 which is obtained from (E.1 by using h 1 = 1, h 2 = 1, h 3 = 1 and using the notation x, y, z in place of x 1, x 2, x 3. If i, j, k denote unit vectors in the eastward, northward and vertical directions, we have for the gradient, divergence and curl ( w i v + j p = i p + j p p, v = u + v + w, ( u w (E.10 (E.11 ( v u. (E.12 E. A. Barnes 5 updated 06:13 on Wednesday 7 th October, 2015

6 Formula (E.9 can also be written in the determinant form i j k. (E.13 u v w Cylindrical coordinates Now let s apply the general formulas to the cylindrical coordinates r, φ, z, where r is the radius, φ is the tangential angle, and z is the vertical distance. The metric expression in cylindrical coordinates is (dl 2 = (dr 2 + (rdφ 2 + (dz 2, (E.14 which is obtained from (E.1 by using h 1 = 1, h 2 = r, h 3 = 1. If i, j, k denote unit vectors in the radial, tangential and vertical directions, we have for the gradient, divergence and curl p = i p r + j p r φ p, (E.15 v = (ru r r ( w i r φ v + j + v r φ + w, ( u w r ( (rv r r (E.16 u. (E.17 r φ Formula (E.14 can also be written in the determinant form i rj k 1. (E.18 r r φ u rv w Spherical coordinates Now let s apply the general formulas to the spherical coordinates λ, φ, r, where λ is the longitude, φ is the latitude, and r is the distance from the center of the earth to the point in question. The metric expression in spherical coordinates is (dl 2 = (r cos φdλ 2 + (rdφ 2 + (dr 2, (E.19 E. A. Barnes 6 updated 06:13 on Wednesday 7 th October, 2015

7 which is obtained from (E.1 by using h 1 = r cos φ, h 2 = r, h 3 = 1. If i, j, k denote unit vectors in the eastward, northward and vertical directions, we have for the gradient, divergence and curl p p = i r cos φ λ + j p r φ p r, (E.20 ( w i r φ (rv + j r r u (v cos φ v = + r cos φ λ r cos φ φ + (r2 w r 2 r, ( ( (ru r r w r cos φ λ v (u cos φ r cos φ λ r cos φ φ (E.21. (E.22 Formula (E.19 can also be written in the determinant form r cos φi rj k 1 r 2. (E.23 cos φ λ φ r ur cos φ vr w E. A. Barnes 7 updated 06:13 on Wednesday 7 th October, 2015