Part I a Few Basic Facts About Atmospheric Fluid Dynamics

Part I

A few basic facts about atmospheric ﬂuid dynamics Outline 1. A bit of history 2. Euler equation of motion for dry atmosphere 3. Various simplifying assumptions 4. Shallow water equations (a) Eulerian formulation (b) Lagrangian formulation (c) Scaling limits and invariants

References: R Salmon “Lectures on geophysical ﬂuid dynamics”, P Lynch “The emergence of numerical weather prediction”, M Cullen “A mathematical theory of large-scale atmosphere/ocean ﬂow”, E Kalnay “Atmospheric modeling, data assimilation and predictability” 1 A bit of history

The mathematical foundation for NWP was laid by Vilhelm Bjerknes and Lewis Fry Richardson

who proposed a mathematical who performed the ﬁrst formulation in terms of seven numerical weather calculation variables (see also Cleveland between 1913 and 1922. Abbe, 1890) and divided the forecast into a diagnostic and prognostic step in 1904

2. The Two Formulations of Fluid Motion

(i) One can either take a ﬁxed (spatial) position with respect to the ﬂuid and formulate the equations of motion from that perspective (Eulerian formulation) or

(ii) one follows the fluid motion, views it as a continuum of in- finitesimal fluid particles and takes a material point of reference to formulate Newtonian type equations of motion (La- grangian formulation).

The time derivatives of a function f in the two formulations are connected by ∂f Df ∂f = = + u · ∇f. ∂τ Dt ∂t Euler’s equations for a rotating ﬂuid in 3D Cartesian geometry are given by Du 1 + fk × u + ∇p + gk = 0, Dt ρ ρt + ∇ · (ρu) = 0, θt + u · ∇θ = 0, 3 where u ∈ R is the velocity (wind) ﬁeld, θ is the potential tem- T perature, k = (0, 0, 1) , f = 2Ω sin φ0 is the Coriolis parameter, g the gravitational constant, and the ideal gas law becomes

R/cp θ = T (p0/p) .

These equations constitute the model for the dry atmosphere (correctly stated by Bjerknes in 1904). 3. Behind the Scenes ...

Eulerian (spatial) Lagrangian (material) coordinates are due to coordinates to

d’Alembert (in 1749) Euler (in 1759) The two people responsible for the confusion are

Lagrange and Dirichlet 4. The continuity equation in the Eulerian and Lagrange frame of reference

Given the initial (material) position x0 of a ﬂuid “parcel”, the ﬂuid “parcel” can be found after time τ at

X(τ, x0) where ∂X = u ∂τ T is the fluid’s velocity field. The “flow” map from x0 = (x0, y0, z0) to X = (X,Y,Z)T is a diffeomorphism for any τ ≥ 0.

Since the mass of a fluid parcel is equal to its density times its volume, we obtain from conservation of mass the relation ∂(X,Y,Z) ρ(0, x0) = ρ(τ, X) . ∂(x0, y0, z0) The integral formulation Z ρ(x, t) = ρ(0, x0) δ(x − X(t, x0)) dA(x0) follows immediately from t = τ and a change of variables using ∂(X,Y,Z) ρ(0, x0) = ρ(τ, X) . (1) ∂(x0, y0, z0) We finally differentiate (1) with respect to τ, to obtain the continuity equation ∂(X,Y,Z) "∂ρ ∂(u, Y, Z) ∂(X, v, Z) ∂(X, Y, w)# 0 = + ρ + ρ + ρ ∂(x0, y0, z0) ∂τ ∂(X,Y,Z) ∂(X,Y,Z) ∂(X,Y,Z) Dρ ∂u ∂v ∂w! = + ρ + + Dt ∂x ∂y ∂z = ρt + ∇ · (ρ u) with u = (u, v, w)T . 5. Various simplifications

If we take Euler’s equations and make the hydrostatic approximation Dw 1∂p 0 = = − − g, Dt ρ ∂z we obtain the primitive equations used by Richardson: Du 1 ∂p − fv = − , Dt ρ ∂x Dw 1 ∂p + fu = − , Dt ρ ∂y 1∂p 0 = − − g, ρ ∂z ρt + ∇ · (ρu) = 0, θt + u · ∇θ = 0, Richardson had the right vision in proposing to discretize the primitive equations by finite difference methods. He even antic- ipated parallel computing. But he used inaccurate initial conditions (data assimilation) which led to disaster ... If we linearize the primitive equations about a motionless state at constant temperature T = 300K, we can determine the wave fre- q 2 2 quency ω as a function of horizontal length scale L = LX + Ly and a discrete set of vertical scales Lz. This is to be compared with the dispersion relation for the un- approximated Euler equations: The Boussinesq approximation relies on the assumption that the relative change in density is very small and needs to be taken into account only in combination with gravitational forces. This assumptions is valid for the ocean but not for the atmosphere. We nevertheless state the equations resulting from the hydrostatic and the Boussinesq approximations: Du ∂φ − fv = − , Dt ∂x Dv ∂φ + fu = − , Dt ∂y ∂φ 0 = − gθ, ∂z ∇ · u = 0, Dθ = 0, Dt We reformulate those equations in isentropic (t, x, y, θ) coordinates. Here θ is the buoyancy/potential temperature and ∂θ/∂z > 0. Since Dθ/Dt = 0, the material time derivative reduces to Df ∂f ∂f = u + v Dt ∂x ∂y in isentropic coordinates. We also introduce B = φ − gzθ and obtain ∂φ ∂φ ∂φ ∂z ∂φ ∂z ∂B |z = | − = | − gθ = | ∂x ∂x θ ∂z ∂x ∂x θ ∂x ∂x θ as well as the hydrostatic relation ∂B = −gz. ∂θ Incompressibility amounts to ∂(x, y, z) = constant. ∂(x0, y0, θ) This yields ∂(x, y, θ) ∂(x, y, z) ∂(x, y, θ) ∂z = = constant. ∂(x0, y0, θ)∂(x, y, θ) ∂(x0, y0, θ)∂θ Taking the material time derivative results finally in the continuity equation Dz ∂u ∂v! θ + z + = 0 Dt θ ∂x ∂y for ∂z z = . θ ∂θ Collecting all equations, we obtain the primitive equations in isentropic coordinates (or generally speaking in a Lagrangian vertical coordinate): Du ∂B − fv = − , Dt ∂x Dv ∂B + fu = − , Dt ∂y ∂B 0 = gz + , ∂θ ∂z ∂ ∂ θ = − (uz ) − (vz ) ∂t ∂x θ ∂y θ with material time derivative D ∂ ∂ ∂ = + u + v Dt ∂t ∂x ∂y and boundary conditions z(t, x, y, θ0) = hS(x, y) at θ0 and B(t, x, y, θ1) = 0 at θ1, θ0 ≤ θ ≤ θ1. 6. The shallow water equations

We now approximate the primitive equations by a single layer model. We set θ0 = 0, θ1 = 1, ∆θ = θ1 − θ0 = 1. Furthermore, we use

z(t, x, y, θ1) = z(t, x, y, θ0) + ∆θ η(t, x, y) = hS(x, y) + η(t, x, y) and

B(t, x, y) = B(t, x, y, θ0) = B(t, x, y, θ1) + g∆θ z(t, x, y, θ1) = g (hS(x, y) + η(t, x, y)) The “layer depth” η ≈ zθ satisﬁes the continuity equation ∂(ηu) ∂(ηv)! ηt = − + . ∂x ∂y The complete set of shallow water equations is now provided by (Eulerian formulation): Du ∂ − fv = −g (hs + η), Dt ∂x Dv ∂ + fu = −g (hs + η), Dt ∂y ∂(ηu) ∂(ηv)! ηt = − + ∂x ∂y with material time derivative D ∂ ∂ ∂ = + u + v . Dt ∂t ∂x ∂y

Here η is the layer depth, hS is the surface orography, g is the gravitational constant, and f is twice the Earth’s angular velocity. The Lagrangian (classical mechanics) formulation in the ﬂuid 2 parcel positions X = (X,Y ) ∈ R becomes ∂2X ∂Y ∂ − f = −g (hs + η), ∂τ 2 ∂τ ∂X ∂2Y ∂X ∂ + f = −g (hs + η) ∂τ 2 ∂τ ∂Y with Z η(x, τ) = η0(x0) δ(x − X(τ, x0)) dA(x0) and initial conditions X(0, x0) = x0, Y (0, x0) = y0 and η0(x) = η(0, x) the initial layer depth. 7. Scaling limits and invariants

We non-dimensionalize the equations by using scaled variables (denoted with a dash) as follows L (x, y) = L(x0, y0), t = t0, u = Uu0, η = Hη0. U Upon introducing U U Ro = (Rossby number), F r = √ (Froude number), fL gH we obtain the scaled shallow water equations (hs = 0) Du0 + Ro−1k × u0 = −F r−2∇η0, Dt0 ∂η0 = −∇ · (η0u0). ∂t0 Let us ﬁrst consider the semi-geostrophic scaling regime where

F r2 = Ro → 0. This regime is observed for global atmospheric ﬂow patterns.

It follows that Du0 Ro + k × u0 = −∇η0 Dt0 and approximate geostrophic wind balance

k × u0 ≈ ∇η0 follows. Taking the curl yields the linear balance relation

∇ × u0 − ∇2η0 ≈ 0. Under the more general scaling limit 1 ε = → 0 Ro−1 + F r−1 we obtain the nonlinear balance relation

0 h 0 0 −1 0 −2 0i ∇ · ut = ∇ · u · ∇u + Ro k × u + F r ∇η ≈ 0.

Nonlinear balance provides a “slow manifold” for small as well as large scale atmospheric ﬂow regimes. It is essential for setting up appropriate initial conditions (Richardson failed to do so) as well as for the stability of numerical schemes (conservation of nonlinear balance).

Note that linear balance is a special case of nonlinear balance. In terms of conserved quantities we have the total energy 1 Z E = η(u2 + v2) + gη2 dA(x) 2 R2 and the material conservation of potential vorticity

vx − uy + f Q = , η i.e., DQ = 0 Dt This in turn implies the conservation of the integrals Z q ξq = η Q dA(x) R2 for any integer q ≥ 1. q = 2 yields conservation of total enstro- phy.