Rossby and Kelvin Waves

Utrecht summer school Physics of the climate system August 15-26, 2016

Rossby and Kelvin waves

Institute for Marine and Atmospheric research Utrecht Utrecht University, Utrecht, The Netherlands Anna von der Heydt Phone: +31-30-2535963 fax: +31-30-2543163 e-mail: [email protected]

2 Contents

1 Introduction 2

2 Overview of wave types 2

3 Analytical preparations 3 3.1 The linearized shallow water equations ...... 3 3.2 The reduced gravity model ...... 4 3.3 Streamfunctions ...... 6

4 The Rossby wave 7 4.1 Structure and propagation characteristics ...... 9 4.2 The propagation mechanism of Rossby waves ...... 12

5 The Kelvin wave 13

1 1 Introduction

In this lecture we will study waves in geophysical fluids. Waves are of great importance in atmospheric and oceanic flows, they transport information through the earth system in the fasted possible way. First, an overview of the various wave types is given. Second, the governing equations of so-called Rossby and Kelvin waves are derived. To this end, the shallow-water equations are linearized and analytical solutions are found. In their linearized form, the equations still represent the main behavior of these wave types. We restrict ourselves to Rossby and Kelvin waves here as the large-scale atmospheric and oceanic disturbances mostly generate these types of waves. Gravity waves and intertial gravity waves may play a role as well, but for practical reasons we leave them out of the discussion here. I found the following textbooks particularly useful: • Holton, J.R. An introduction to dynamic meteorology, Academic Press, Orlando, 1992. • Cushman-Roisin, B. Introduction to Geophyiscal Fluid Dynamics, Prentice- Hall, Englewood Cliffs, New Jersey, 1994. • Pedloski, J. Waves in the ocean and atmosphere, Introduction to wave dynamics, Springer, New York, 2003. • And for those who would like to see real Rossby waves in the ocean: D. B. Chelton and M. G. Schlax, Global Obeservations of Oceanic Rossby waves, Science 272, pp. 234-238 (1996)

2 Overview of wave types

For all waves two balancing components are required. Inertia is relevant for every wave type, but the balancing component varies. A list of the main wave types and their common names are listed in Table 1. Waves that make use of the (small) compressibility of air or water are acoustic or sound waves. However, acoustic waves have a too high wave speed to be relevant for geophysical flows. A second group of wave types are those who exist in barotropic flows, i.e. flows with uniform density. The incompressibility principle implies that the flow is non-divergent. Two steering forces are gravity and Coriolis. The simplest waves of this type are Kelvin waves, which will be analyzed below. Kelvin waves need a boundary, like the shore for the oceans or, strangely enough, the equator. Without boundaries, these type of waves are Poincaréwaves. In fact, tsunamis are short Poincaréwaves breaking on the coast, for which Coriolis forces are less important. Kelvin and Poincaréwaves travel relatively fast. Rossby waves travel much slower, because they react on variations in the Coriolis force with the latitude. They have similarities with topographic waves, although these waves need a background flow.

2 Table 1: Overview of wave types

Name Balancing term

Essentially compressible Acoustic or sound waves Pressure variations

Incompressible, barotropic Kelvin waves Gravity and Coriolis along a ﬂuid boundary Poincar´ewaves Gravity and Coriolis Rossby waves (or planetary waves) Coriolis parameter variations Topograpic waves Flow depth variations

Incompressible, baroclinic Internal (gravity / Buoyancy) waves Density diﬀerences

In reality, the atmosphere and the ocean have a density that changes with height. That adds one other type of waves, namely internal waves. These waves are the least visible type of waves because these are completely inside the ﬂuid and often are conﬁned to narrow zones.

3 Analytical preparations 3.1 The linearized shallow water equations Here, we restrict ourselves to linear waves because they describe the dominant wave phenomena quite well. For simplicity, we start with a barotropic, one-layer system, thus the shallow water equations that have been derived in a previous lecture. We linearize these equations in order to ﬁnd plain wave solutions. This simplicity is searched so that the physical mechanisms behind the wave dynamics can be understood. The shallow water equations are

∂u ∂u ∂u ∂ξ + u + v − fv = −g ∂t ∂x ∂y ∂x ∂v ∂v ∂v ∂ξ + u + v + fu = −g ∂t ∂x ∂y ∂y ∂h ∂uh ∂vh + + = 0 ∂t ∂x ∂y h = ξ + H in which we recall that ξ is the surface elevation, i.e. the elevation of the upper surface of the ﬂuid.

3 We linearize the equations around a state of rest, i.e. a state with zero velocity, and zero surface elevation. The idea of linearization is that all variables (u, v, ξ) are assumed to have inﬁnitesimal small amplitude. This means that terms linearly dependent on these variables have very small magnitude. How- ever, nonlinear terms, like the advection terms, e.g. uux have a magnitude that is even much and much smaller. So small in fact, that we can neglect them. We then arrive at the following set of equations: ∂u ∂ξ − fv = −g ∂t ∂x ∂v ∂ξ + fu = −g ∂t ∂y ∂h ∂u ∂v + H( + ) = 0 ∂t ∂x ∂y h = ξ + H These equations are the basis of what is to follow.

3.2 The reduced gravity model There are serious restrictions in applying the shallow-water equations to the real atmosphere and the real ocean. This is because fluids are treated as homogeneous in the vertical, i.e. the density is assumed to be the same every where. In reality, stratification is one the key steering factors of oceanographic and atmospheric flows and cannot easily be neglected. Isn’t it possible to build in some variation of density with height, so some stratification? The answer is yes. One can put a pile of shallow-water models on top of each other and connect them properly. The simplest of those models is the reduced gravity model (Figure 1). In the reduced gravity model the fluid is assumed to consist of two layers of different density, in which the lower layer fluid is at rest. To keep the lower layer at rest the horizontal pressure gradients have to be zero there. This introduces a special relation between the upper surface of the fluid and the interface between the two layers. It is shown below that the reduced gravity model has the same form as the shallow-water model, but the terms have a slightly different meaning. The reduced gravity model presented here is set up to represent the ocean, but with some reformulation such model can also represent the atmosphere. Then, density must be replaced by potential density and depth by height. Fur- thermore, the model can be extended to a multilayer reduced gravity model, in which still the lowest layer is at rest. We first integrate the pressure from the top of the fluid to somewhere in the lower layer, to level z, say. Using the hydrostatic equation we find: Z ξ Z ξ pz dz = − gρ dz z z The right-hand side of this equation can be evaluated by introducing the elevation of the interface between the two layers, relative to a state of rest, as η. We

4 ξ

Upper layer ρ1

Lower layer ρ2

Figure 1: The reduced-gravity model, with surface elevation ξ and interface elevation η. Note that η is deﬁned positive downward. choose η as positive downward, as depicted in Figure 1. We then ﬁnd:

Z ξ Z −(H+η) Z ξ − gρ dz = − gρ2 dz − gρ1 dz z z −(H+η)

= −g(ξ + H + η)ρ1 − g(−(H + η) − z)ρ2

The left-hand side becomes: Z ξ pz dz = p(ξ) − p(z) = −p(z) z where we use that the pressure above the ﬂuid p(ξ) is negligible. Hence the pressure at level z is given by:

p(z) = g(ξ + H + η)ρ1 − g(z + H + η)ρ2

Our assumption is that the lower layer fluid is at rest, or that the horizontal pressure gradients are zero there. Taking the horizontal gradient of p(z) leads to: ∇p = gρ1∇ξ − g(ρ2 − ρ1)∇η = 0 so that we find: gξ = g0η 0 0 in which g is the reduced gravity acceleration given by g = g(ρ2 − ρ1)/ρ1. Since the density difference between the layers is usually much smaller than the density itself we find that η >> ξ. This allows us to neglect ξ compared to η in the continuity equation, so that the set of equations describing the motion in the first layer becomes:

5 ∂u ∂η − fv = −g0 ∂t ∂x ∂v ∂η + fu = −g0 (1) ∂t ∂y ∂η ∂u ∂v + H( + ) = 0 ∂t ∂x ∂y where we used that ht = ηt, because H is constant. Notice that these equations for the reduced gravity model diﬀer from those of the shallow-water equations in that g0 replaces g, and η replaces ξ.

3.3 Streamfunctions A useful property in quasi-geostrophic analysis is the streamfunction ψ captur- ing the flow potential. It has some similarities with the surface interface height η, because η can also been seen as a flow potential. In the definition of ψ, we use that the first order momentum balance is geostrophic: ∂η −f v = −g0 (2) 0 ∂x ∂η f u = −g0 0 ∂y If we eliminate the pressure gradient terms by cross differentiation (i.e. take the zonal derivative of the second and subtract the meridional derivative of the first) we find: f0(ux + vy) = 0 So, the horizontal divergence of the horizontal velocities is zero, as we have found before. However, this relation also allows us to introduce a streamfunction ψ as:

u = −ψy (3)

v = ψx because now f0(ux + vy) = f0(−ψxy + ψxy) = 0 for all forms of ψ. By introducing the streamfunction we make sure that we fulfill the continuity equation automatically. Another good thing about introducing the streamfunction is that it allows us to reduce the number of unknown variables from 3 (u, v, and η) to 2. In fact, also η can be written in terms of the streamfunction. From the definition of the streamfunction, ψ is defined up to a constant, which might be time dependent. If we relate the definition of the streamfunction (3) with the geostrophic relations (2) we find f η = 0 ψ + C(t) (4) g0

6 Figure 2: Atmospheric Rossby waves encircle the north pole. in which C(t) a time dependent constant. From now on we choose C(t) = 0. (In fact, C(t) has to do with mass conservation, and in a closed domain this becomes important. Here, we only treat waves on an inﬁnite plane, and it can be shown that C has to be constant, and any constant will do.)

4 The Rossby wave

The Rossby wave is interesting because its driving mechanism is the planetary vorticity ﬁeld of the rotating earth. There are several starting points for an analysis of Rossby waves, here we start with the reduced gravity shallow water equations (Equations 1). The driving mechanism of Rossby waves is the interaction of the ﬂow with meridional variations of the Coriolis parameter f. This parameter f depends on the latitude with f = 2Ω sin ϕ in which Ω and ϕ are the earth’s angular velocity and latitude on the earth, respectively. If the coordinate y is oriented northward and is measured from a reference latitude ϕ0, for example the latitude in the middle of the wave under consideration, then ϕ = ϕ0 +y/a, where a is the earth’s radius (6371 km). Using

7 Taylor expansion, we ﬁnd y f = 2Ω sin ϕ + 2Ω cos ϕ + ... 0 a 0 By approximation, this is

f ≈ f0 + β0y, (5) in which f0 = 2Ω sin ϕ0 is the reference Coriolis parameter and β0 = 2Ω/a cos ϕ0 −5 −1 is the beta parameter. Typical mid-latitude values are f0 = 8 × 10 s and −11 −1 −1 β0 = 2 × 10 m s . If this approximation of f is applied in a Cartesian framework, this framework is called the f-plane if the beta term is neglected, and the beta plane if the beta term is retained. When we apply the expansion of f in Equations (1), we find ∂u ∂η − (f + β y)v = −g0 (6) ∂t 0 0 ∂x ∂v ∂η + (f + β y)u = −g0 (7) ∂t 0 0 ∂y ∂η ∂u ∂v + H( + ) = 0. (8) ∂t ∂x ∂y These equations, however, contain first and second other terms. The first or- 0 der terms (f0, g and H terms) comprise the f-plane geostrophic dynamics. The smaller ones (time derivatives and β0 terms), are perturbations on this geostrophic flow, but govern the Rossby wave evolution. In order to solve these equations, we use this first order geostrophic balance to eliminate u and v in the smaller terms of Equations (6) and (7). To recap, the first order geostrophic balance reads g0 ∂η u ' − fo ∂y g0 ∂η v ' . fo ∂x When we use this approximation for the momentum equations, we find

0 2 0 g ∂ η β0g ∂η 0 ∂η − − f0v − y = −g f0 ∂y∂t f0 ∂x ∂x 0 2 0 g ∂ η β0g ∂η 0 ∂η + + f0u − y = −g , f0 ∂x∂t f0 ∂y ∂y which can easily be solved to

0 0 2 0 g ∂η g ∂ η β0g ∂η u = − − 2 + 2 y f0 ∂y f0 ∂x∂t f0 ∂y 0 0 2 0 g ∂η g ∂ η β0g ∂η v = + − 2 − 2 y . f0 ∂x f0 ∂y∂t f0 ∂x

8 The first term of both equations are the geostrophic components, the next terms are called ageostrophic. These terms provide a first order behavior to perturbations on the geostrophic flow. Final substitution in the continuity equation (8) leads to a single equation for the surface interface displacement:

1 ∂η ∂ 2 ∂η 2 − ∇ η − β0 = 0, (9) Rd ∂t ∂t ∂x √ 2 0 where ∇ is the two-dimensional Laplace operator and Rd = g H/f0 is the Rossby deformation radius. The Rossby radius for the reduced gravity model is also called the internal Rossby radius of deformation, or sometimes the baroclinic one. The internal Rossby radius of deformation is about 1000 km for the atmosphere and 30 to 100 km for the ocean. Reason for this nomenclature is that it takes the internal structure of the fluid into account. As you will see later, and will find also in other lectures, the Rossby radius of deformation is crucial in our understanding of geophysical fluid flows. Interestingly, for the one layer model we find the same equation, but now the Rossby-radius of deformation is given by: √ gH Rd = f0

The Rossby radius of deformation for the one-layer shallow-water equations is also called the external Rossby radius of deformation, or the barotropic Rossby radius of deformation. For atmosphere and ocean its numerical value is a few thousand km. Here, we derive the quasi-geostrophic equation on a beta plane as a function of η. This equation can be expressed as a function of the streamfunction ψ. This equation,

∂ 2 1 ∂ψ ∇ ψ − 2 ψ + β = 0 (10) ∂t Rd ∂x is the quasi-geostrophic vorticity equation for the reduced-gravity model.

4.1 Structure and propagation characteristics We study wave-like solutions of the quasi-geostrophic surface displacement equation of the form: η(t, x, y) = A cos(kx + ly − ωt) in which k = 2π/λ is the zonal wavenumber, l = 2π/µ is the meridional wavenumber, and ω is the angular frequency, also written as ω = 2π/P in which P is the period of the wave, while λ and µ are the wavelength of the

9 wave in zonal and meridional direction. This so-called plane wave is inserted in Equation (9) or (10), leading to: 2 2 1 −ω k + l + 2 A sin(kx + ly − ωt) − βkA sin(kx + ly − ωt) = 0 Rd This relation is true for all x, y, and t when the factor in front of the sin(..) is zero, so: 2 2 1 −ω k + l + 2 − βk = 0 Rd or βk ω = − (11) 2 2 1 k + l + 2 Rd This relation that ensures that the plane wave is a solution to the quasi- geostrophic surface interface equation is called the dispersion relation. It states how angular frequency ω and the two horizontal wavenumbers k and l are related. In Figure 3(a) the dispersion relation is drawn for zero meridional wavenumber. From the dispersion relation the phase speed of the wave can be found, so the speed at which the wave crests move. The phase speed in the zonal direction is given by the zonal wavelength divided by the wave period:

λ ω β c(x) = = = − 2 2 1 P k k + l + 2 Rd

Note that the zonal phase velocity is always negative! This result is one of the most important results on Rossby waves: all Rossby waves have a westward phase velocity. The meridional phase speed is found as:

(y) µ ω βk c = = = − P l 2 2 1 l k + l + 2 Rd which has no preferred direction. Another interesting quantity is the velocity of the wave energy ﬂux. This velocity is given by the group velocity, and its zonal component is deﬁned as

2 2 1 ∂ω β −k + l + R2 Cg(x) = = − d 2 ∂k 2 2 1 k + l + 2 Rd

This relation is interesting because it shows that waves with large wave lengths, so-called long waves, which have relatively small wavenumbers k and l, have westward group velocity. Hence they can only transport energy westward! The

10 |ω|

1/Rd κ

(a) (b)

Figure 3: a) The dispersion relation of free Rossby waves for zero meridional wavenumber. b) The group speed of free Rossby waves for zero meridional wavenumber. fastest waves are those with the longest wavelenghts, and the maximum is found 2 as −βRd. This is illustrated in Figure 3(a), which shows the dispersion relation of free Rossby waves with zero meridional wavenumber l = 0. The group velocity, which is the derivative of the curve, is zero at k = 1/Rd, negative for longer waves k < 1/Rd, indicating westward energy transport, and ﬁnally positive for short waves k > 1/Rd, indicating eastward energy transport. Figure 3(b) shows the groups speeds for l = 0. This is one example of the importance of the Rossby radius of deformation. Note that the phase velocity, which is the velocity of the wave crests, can have a diﬀerent direction than the group velocity, which is the transport direction of the wave energy! The meridional component of the group velocity is given by:

∂ω 2klβ Cg(y) = = 2 ∂l 2 2 1 k + l + 2 Rd

Interestingly, the ratio of meridional phase and group velocity is always negative, so that they always have opposite direction! The components of group velocities are the derivatives of the angular frequency of the waves to the diﬀerent wavenumbers. When the group velocities become zero, the angular frequency is maximal. When the zonal group velocity is zero this maximum is found when

2 2 2 k = l + (1/Rd)

11 and ωmax becomes: p 2 2 β l + (1/Rd) ωmax = − 2 2 2(l + (1/Rd) ) So, depending on the meridional wavenumber and the background stratiﬁcation, certain Rossby waves cannot exist. For instance, for purely zonal waves with l = 0 the maximum frequency is

βR ω = − d max 2

For the ocean this limit is quite restricting due to the relatively small baroclinic Rossby radius of deformation. No baroclinic zonal Rossby waves with periods shorter than 90 days are possible poleward of 15o latitude, and the 180-day period boundary lies at 30o latitude!

4.2 The propagation mechanism of Rossby waves Let us now try to understand why the wave crests do move westward. For very short Rossby waves the wavenumbers are much larger than the inverse of the Rossby deformation radius squared. The potential vorticity balance (10) than reduces to ∂∇2ψ + βψ = 0 ∂t x or, in terms of relative and planetary vorticity: d (ζ + f) = 0 (12) dt

2 where we used βψx = βv = df/dt and ∇ ψ = ζ. We see that the stretching terms, related to Rd, can be neglected. In Figure 4 an illustration is given of the propagation mechanism. Suppose a ﬂuid parcel has moved pole-ward compared to its environment. The magnitude of its planetary vorticity has increased, so it has obtained an anticyclonic relative vorticity, according to (12). Because of this, it will act as a vortex for surrounding parcels. Its sense of rotation is such that parcels west of it are transported northward, gaining planetary vorticity. This gain is compensated for, according to (12), by an anticyclonic relative vorticity change of these parcels. So, the anticyclonic vorticity of the original parcel is transported westward. The parcels that have gained anticyclonic vorticity will behave like vortices, and inﬂuence parcels further west, leading to westward propagation of the wave crests. The motion that they induce on their eastward side, where the original parcel lies, is directed southward, pushing the original parcel back to its initial position: the wave crest has passed by. For very long waves the vorticity balance (10) reduces to: ∂ψ 1 ∂ψ β − 2 = 0 ∂x Rd ∂t

12 vortex circulation

t=0

Induced motion Induced motion

t>0 Induced circulation

Figure 4: Propagation mechanism for short waves. or, in terms of vorticity itself:

d f η f − 0 = 0 (13) dt H in which we used the relation between the streamfunction and the interface elevation, equation (4). Now the balance is between planetary vorticity and stretching, and relative vorticity can be neglected. A hump at the surface will be accompanied by geostrophic velocities along contours of equal height, according to geostrophy. At the westward side the motion is pole ward, and at the eastward side the motion is equator ward. The pole ward motion leads to an increase in the magnitude of the planetary vorticity, and has to be accompanied by an increase in magnitude of the stretching term. So, the surface elevation will increase. On the eastward side the induced equator ward motion leads to a decrease in surface elevation. The net result is a westward motion of the hump. For wavenumbers in between these two extremes both mechanisms are at work. Note again the importance of the Rossby radius of deformation in the physical mechanism at work.

5 The Kelvin wave

Another kind of wave of major importance is the Kelvin wave. This kind of wave requires the support of a lateral boundary. The wave is not geostrophic, so we start from the shallow-water equations. In this section we only treat the one-layer case; for Kelvin waves in stratiﬁed ﬂuids reference is made to the

13 literature. The linearized shallow-water equations for a plane bottom read: ∂u ∂ξ − fv = −g ∂t ∂x ∂v ∂ξ + fu = −g (14) ∂t ∂y ∂ξ ∂u ∂v + H( + ) = 0 ∂t ∂x ∂y Suppose now that a meridional ridge is present at x = 0, with the ﬂuid eastward of the ridge. Because the zonal velocity has to be zero at the meridional ridge, we search for solutions in which u = 0 everywhere. The equations then reduce to: ∂ξ −fv = −g ∂x ∂v ∂ξ = −g (15) ∂t ∂y ∂ξ ∂v + H = 0 ∂t ∂y Note that the assumption on the zonal velocity makes the wave geostrophic in the zonal direction, as the ﬁrst of these equations shows. We assume a wave traveling in the meridional direction:

i(ly−ωt) v = v0(x)e (16) i(ly−ωt) ξ = ξ0(x)e in which v0(x) and ξ0(x) are complex amplitudes that depend on the zonal coordinate. Using this in the governing equations we ﬁnd: ∂ξ −fv = −g 0 0 ∂x −ωv0 = −glξ0

−ωξ0 + Hlv0 = 0 The latter two homogeneous coupled algebraic equations have nontrivial solutions if the determinant is zero:

ω2 − gHl2 = 0 leading to two possible dispersion relations:

ω = ±pgHl

The sign is determined by the geostrophic relation in zonal direction. The ﬁrst two governing equations can be used to eliminate v0 to ﬁnd: ∂ξ fl 0 = ξ ∂x ω 0

14 which can be solved as: (fl/ω)x ξ0 = Ae The positive sign in the dispersion relation gives exponentially growing solutions in x so they have to be discarded as unphysical. (At least in the Northern Hemisphere. In the Southern Hemisphere the negative sign gives unphysical solutions.) This leaves us with the dispersion relation

ω = −pgHl (17) and solution

ξ = A cos(ly − ωt)e−x/Rd r g v = −A cos(ly − ωt)e−x/Rd H in√ which Rd is the now well-known Rossby deformation radius, given by Rd = gH/f. Interesting about this wave is that it can only travel southward along a ridge on its west. This is a general feature of Kelvin waves: they travel along ridges to their right. On the Southern Hemisphere Kelvin waves travel along ridges to their left. This preference of travel direction must be due to the Coriolis force because that force introduces directions in the problem. Indeed, the wave is in geostrophic balance in the zonal direction, so an eastward directed pressure force (related to a positive surface elevation) is balanced by a westward directed Coriolis force, and the other way around. This means that when the sea-surface elevation is positive the parcels have to move southward to generate that Coriolis force, and northward when the sea-surface elevation is negative. This is consistent with the wave dynamics when the wave crests move southward. The phase velocity of the wave follows from the dispersion relation as:

c(y) = −pgH

Since this velocity does not depend on l the wave is nondispersive. The group velocity is in that case equal to the phase velocity:

(y) p cg = − gH

It is left as an exercise for the reader to show that Kelvin wave solutions in a reduced gravity model are simply obtained from the solution presented above by replacing g with g0 and by interpreting ξ as the interface deviation. A special kind of Kelvin waves play an important role in the equatorial dynamics. Here no solid boundary is present to lean against, but a Kelvin wave in the Northern Hemisphere ’leans’ against a similar one on the Southern Hemisphere. It can be shown that this conﬁguration has to move eastward. It plays a dominant role in e.g. the El Nino phenomenon, but unfortunately we cannot treat these waves in this short lecture. Use the literature at the beginning of this chapter to learn more about these exciting waves!

15 Acknowledgments I thank Peter Jan van Leeuwen and Willem Jan van de Berg for providing me with their material for preparing this lecture.