ATMS 310 Math Review

Differential Calculus

dy The first derivative of y = f(x) = = f `(x) dx 2 xd The second derivative of y = f(x) = = f ``(x) dy2 Some differentiation rules: dyn = nyn-1 dx

()uvd dv du = u( ) + v( ) dx dx dx

⎛ u ⎞ ⎛ du dv ⎞ d⎜ ⎟ ⎜v − u ⎟ v dx dx ⎝ ⎠ = ⎝ ⎠ dx v2

dy ⎛ dy dz ⎞ Chain Rule If y = f(z) and z = f(x), = ⎜ *⎟ dx ⎝ dz dx ⎠

Total vs. Partial Derivatives

The rate of change of something following a fluid element (e.g. change in temperature in the atmosphere) is called the Langrangian rate of change:

d / dt or D / Dt

The rate of change of something at a fixed point is called the Eulerian (oy – layer’ – ian) rate of change:

∂/∂t, ∂/∂x, ∂/∂y, ∂/∂z

The total rate of change of a variable is the sum of the local rate of change plus the 3- dimensional advection of that variable:

d() ∂( ) ∂( ) ∂( ) ∂( ) = + u + v + w ∂tdt ∂x ∂y ∂z (1) (2) (3) (4)

Where (1) is the Eulerian rate of change, and terms (2) – (4) is the 3-dimensional advection of the variable (u = zonal wind, v = meridional wind, w = vertical wind)

Vectors

A scalar only has a magnitude (e.g. temperature). A vector has magnitude and direction (e.g. wind).

Vectors are usually represented in bold font. The wind vector is specified as V or r sometimes as V i, j, and k known as unit vectors. They have a magnitude of 1 and point in the x (i), y (j), and z (k) directions. See the figure below.

The Total Wind vector V = iu + jv + kw , where u,v, and w are the scalar components of the zonal, meridional, and vertical wind.

Vector Mathematics

Vector Addition/Subtraction – Simply add/subtract the scalars of each component together.

V1+V2 = (u1+u2)i + (v1+v2)j + (w1+w2)k

Graphically, draw the first vector. Place the second vector’s tail (non-arrow end) on the first vector’s head (arrow end). Draw a new vector from the tail of the first to the head of the second vector to find the sum:

Vector Multiplication – There are two forms of vector multiplication – the dot product and the cross product. The dot product is defined as the product of the magnitudes of the vectors and the cosine of the smallest angle between them. It results in a scalar quantity:

V1•V2 = V1V2cos(θ) = u1u2+v1v2+w1w2

Note that i•i = j•j = k•k = 1 and that the dot products between any two different unit vectors = 0.

The cross product results in a third vector that points in the direction given by the “right hand” rule. It has a magnitude given by:

|| V1 x V2 || = V1V2sin(θ)

Where || || denotes “magnitude of”, ‘x’ means cross product, and θ is the angle measured clockwise from V1 to V2. The resultant vector is perpendicular to the first two. The total cross product can be calculated as:

V1 x V2 = i(v1w2 – v2w1)+j(u2w1-u1w2)+k(u1v2-u2v1)

It can also be expressed as a determinant:

kji

V1 x V2 = wvu 111

wvu 222 Note that i x i = j x j = k x k = 0 since θ = 0 and sin(0) = 0 Note also that i x j = k, j x k = i, and k x i = j

The cross product is typically used in meteorology where there is rotation involved (e.g. vorticity).

The “Del” or Differential Operator

∂ ∂ v ∂ =∇ i + vv j + k ∂x ∂y ∂z

If the del operator is multiplied by a scalar (e.g. ‘a’), it is known as the “gradient of a”.

∂a ∂a v ∂a =∇ ia + vv j + k ∂x ∂y ∂z

If we dot multiply a velocity vector (V) with the del, it yields the divergence of V.

∂u ∂v ∂w ∇ •V= + + ∂x ∂y ∂z

Frequently we dot the 2-dimensional (horizontal) wind with the del operator to yield the horizontal wind divergence.

If we multiply a velocity vector (V) by cross product with the del, it yields the curl.

v v v kji ∂ ∂ ∂ ⎛ ∂w ∂v ⎞ ⎛ ∂u ∂w ⎞ ⎛ ∂v ∂u ⎞ ∇ x V = = i⎜ − ⎟ +j⎜ − ⎟ +k⎜ − ⎟ ∂ ∂ ∂zyx ⎝ ∂y ∂z ⎠ ⎝ ∂z ∂x ⎠ ⎝ ∂x ∂y ⎠ wvu (1) (2) (3)

Term (3) is the rotation in the horizontal plane. This is the common expression for relative vorticity (ζ) in Cartesian coordinates. We can arrive at this expression by taking the dot product of the 3-D wind cross product with the k unit vector:

∂v ∂u ζ = k•(∇ x V) = − ∂x ∂y

If we dot multiply the del operator with itself, we have the Laplacian operator (∇2 ):

∇=∇•∇ 2

If we apply the Laplacian to a scalar, we get:

∂2a ∂2a ∂2a 2aa =∇=∇•∇ + + ∂x2 ∂y2 ∂z 2

Finally, if we dot multiply the gradient of a scalar (e.g. T for temperature) with the wind vector, we get the advection of temperature by the wind:

∂T ∂T ∂T V•∇ T = u + v + w ∂x ∂y ∂z

Euler’s Relation shows how a total derivative can be broken down into a local rate of change + the advection of a quantity. For example, the total derivative of temperature (T) can be written as:

dT ∂T ∂T ∂T ∂T = + u + v + w dt ∂t ∂x ∂y ∂z

Substituting from the advection relationship above, we get:

dT ∂T = + V•∇T dt ∂t