Dynamics

Equations

local rate of change in u

du/dt

advection in u

u*du/dx + v*du/dy + w*du/dz

pressure gradient force in u

-1/rho * dp/dx

coriolis force in u

2*O*v*sin(phi) - 2*O*w*cos(phi)

curvature in u

u*v*tan(phi)/Re - u*w/Re

viscous force in u

c*del²u

local rate of change in v

dv/dt

advection in v

u*dv/dx + v*dv/dy + w*dv/dz

pressure gradient force in v

-1/rho *drho/dy

coriolis force in v

-2*O*u*sin(phi)

curvature in v

-u²/Re*tan(phi)-w*v/Re

viscous force in v

c*del²v

local rate of change in w

dw/dt

advection in w

u*dw/dx + v*dw/dy + w*dw/dz

pressure gradient force in w

-1/rho*dp/dz

gravity in w

-g

coriolis force in w

2*o*u*cos(phi)

curvature in w

u²+v²/Re

viscous force in w

c*del²w

del operator

d/dx*i + d/dy*j + d/dz*k

laplacian

del² = d²/dx² + d²/dy² + d²/dz²

divergence

del . vector

divergence of wind

du/dx + dv/dy + dw/dz

advection of temperature

U . del(T)

advection of temperature

u*dt/dx + v*dT/dy + w*dT/dz

coriolis parameter f

2*O*sin(phi)

omega vector

O*cos(phi)*j + O*sin(phi)*k

change in wind over time due to coriolis

-2*omega vector X wind vector

change in horizontal wind over time due to coriolis

(f*v)*i - (f*u)*j

largest terms in horizontal momentum

horizontal pressure gradient force, horizontal coriolis force

geostrophic wind vg

1/f *dp/dx

geostrophic wind ug

-1/f *dp/dy

largest terms in vertical momentum

vertical pressure gradient force, gravity

hydrostatic equation

dp/dz = -rho*g

equation of state

p = rho*R*T

scale height H

R*T/g

vertical thickness ZT

R/g0*<avg layer T>*ln[p(z1)/p(z2)]

change in geopotential wrt x

1/rho*dp/dx

continuity equation drho/dt

-[d(rho*u)/dx + d(pho*v)dy + d(rho*w)dz]

continuity equation drho/dt

-del . (rho*U)

lagrangian form of continuity equation

1/rho * Drho/Dt

rossby number R0

Du/Dt / f*u

rossby number for pure geostrophy

R0 = 0

rossby number to assume geostrophy

<= 10^-1

rossby number definition

the ratio of the inertial acceleration to the coriolis acceleration

synoptically scaled continuity equation

w/rho0 * drho0/dz + del . U

mass divergence on synpotic scale

del . (rho0*U) = 0

atmosphere is non-divergent

on synpotic scale with absence of vertical motion

first law of thermodynamics J

De/Dt + p*Dalpha/Dt = J

specific heat at constant volume cv

(Dq/Dt)v = De/DT

first law of thermodynamics cv

cv*DT/Dt + p*Dalpha/Dt = J

specific heat at constant pressure cp

(Dq/DT)p = cv + R

first law of thermodynamics cp

cp*DT/Dt - alpha*Dp/Dt = J

entropy is achieved

only through heat transfer

first law of thermodynamics entropy S

cp*Dln(T)/Dt - R*Dln(p)/Dt = DS/Dt

poisson’s equation T0

theta = T(p00/p)^(R/cp)

first law of thermodynamics potential temperature theta

cp*Dln(theta)/Dt = DS/Dt

buoyancy force/acceleration

(rho0 - rho)/rho * g

buoyancy force/acceleration potential temperature theta

Fvert/m = (theta - theta0)/theta0 * g

D^2(deltaz)/Dt^2

-g*dln(theta0)/dz * deltaz

statically stable

dtheta0/dz > 0

statically neutral

dtheta0/dz = 0

statically unstable

dtheta0/dz < 0

incompressible fluid

changes in density are independent of changes in pressure, equation of state is rho = f(T,k), changes in density can only occure as a result of temperature changes

first law of thermodynamics incompressible atmosphere

cp*DT/Dt = J

incompressible atmosphere

synoptically scaled atmosphere that is adiabatic with no vertical motion

vertical component of velocity in pressure coordinates omega

Dp/Dt = omega

total derivative in pressure coordinates D/Dt

d/dt + u*d/dx + v*d/dy + omega*d/dp

synoptically scaled prognostic momentum equation height coordinates in u

f*v - 1/rho*dp/dx

synoptically scaled prognostic momentum equation height coordinates in v

-f*u - 1/rho*dp/dy

geostrophic wind ug pressure coordinates

-1/f*dgeopotential/dy

geostrophic wind vg pressure coordinates

1/f*dgeopotential/dx