Hipótesis específica 3

Earth, this allows us to use the cardinal points to describe the system. The rotation vectorΩ always goes through the North and South poles, the North pole is defined as the one about which rotation is anti-clockwise, as on Earth. East is prograde with respect toΩ, in the direction of increasing longitude. The term prograde will be used to describe motion in the direction of rotation, retrograde in the opposite direction. All this is to say, when looking at maps of exoplanets the rotational asymmetry is the same as on Earth and so direct comparisons can be made.

Unless otherwise stated, the atmosphere will be dry with an Earth-like composition, using the values of thermodynamic properties stated in the table of constants. Gravity and planetary radius will be kept constant at Earth’s values. For simplicity, higher moments of orbit – obliquity, eccentricity, precession etc. – are all zero so that orbits are perfectly circular. Therefore all modelled planets have a constant orbital rate,Γ , the substellar point is always on the equator, and the strength of stellar irradiation is constant in time.

Due to the very high aspect-ratio of terrestrial planetary atmospheres, there will often be a scale separation between horizontal motion and vertical motion. Unless stated otherwise, the Laplacian operator applied to a three-dimensional field will act in the horizontal only ∇2_{ψ =} ∂ 2_ψ ∂ x2 + ∂2_ψ ∂ y2, (1.5)

with equivalent metric terms included when working with a spherical geometry. The divergence operator,∇·, will apply to all dimensions of the vector field. If we wish

to apply it along a specific plane, it will be denoted with the dimension omitted as subscript. For example, divergence in the horizontal is given by

∇z· u =

∂ u ∂ x +

∂ v

∂ y. (1.6)

1.4

An outline of what follows

While it would be good to immediately get into the problem at hand, it is helpful to define the system we will be working with and the models of the atmosphere that we

12 INTRODUCTION

will rely on for the remainder of this thesis. Chapter 2 introduces the equations of motion of the atmosphere. Starting from the most complete set of equations — the Navier-Stokes equations of fluid motion and thermodynamics — we derive a series of approximations to more easily identify relationships of cause and effect, hopefully a greater gain in understanding offsetting a smaller loss in fidelity.

In chapter 3 we provide an overview of exoplanets and previous work done on the modelling of exoplanetary atmospheres — what we know and what we do not. We will discuss detection methods, specifically transit detection, and the thermal phase curvethat is created by continuous monitoring of an exoplanet orbiting a distant star. We identify several key features of the exoplanet atmospheric circulation, especially where they differ from our intuition of the weather on Earth.

It is common to assume that an exoplanet is tidally locked, its rotation rate equal to orbital rate, as the moon is to Earth. Due to the strong gravitational tidal forces acting over small distances tidal locking is a likely state for planets close-in to their host star, but not universal, as is clear from our Solar System where none of the planets are tidally locked1. In chapter 4 we investigate the relationship between the offset in the observed thermal phase curve and the rotation and orbital rate of a planet. Using a shallow water approximation of the atmosphere, we show that an offset in the phase curve of an observed exoplanet could be a result of asynchronous rotation, with both prograde and retrograde offsets feasible when only these two orbital parameters are varied.

Shallow water theory does not tell the whole story of atmospheric circulation, and so in chapter 5 we extend our investigation into three dimensions. The introduction of vertical stratification allows for more complex dynamics; where our shallow water solutions converged to a steady state, in the stratified system inherent instabilities mean that we only ever obtain a statistically steady state, baroclinic instability and turbulent motion produce synoptic scale variation but can also provide a source of momentum for the mean flow, generating coherent jets. We show that the formation of a superrotational jet, a robust feature of many “fully-featured” general circulation

1_{Mercury is technically tidally locked, but due to its high orbital eccentricity it is locked in a 3:2}

1.4 AN OUTLINE OF WHAT FOLLOWS 13

models of tidally locked exoplanets, is modulated by the presence and direction of a diurnal cycle. The results of chapter 4 are shown to qualitatively agree with those of the stratified model; we discuss the differences between the two and the utility of the shallow water model in capturing the large-scale dynamics of a diurnally forced planet.

We have an interest in exploring how the results of our exoplanet studies could be applied to further understanding of our own atmosphere here on Earth. Chapter 6 introduces the problem of explaining the underlying mechanism that drives the Madden-Julian Oscillation, a perturbation in the largely homogeneous weather in the tropics. Propagating slowly eastward, it is the primary source of variability in tropical weather but it is not well understood – operational weather forecasting models have difficulty in reproducing both its intensity and propagation speed, and the physical driving forces of the MJO are still unknown. Again using a shallow water formulation, this time with a simple parameterisation of the effects of water vapour advection and condensation, we show the feedback of latent heat release onto the atmosphere can induce a self-perpetuating response and slowly propagating perturbation, analogous to the thermal response leading the diurnal cycle on an exoplanet.

The vast proportion of the numerical studies presented in this thesis were performed using Isca, an open-source fork of the Princeton GFDL FMS flexible modelling system. Isca was developed primarily here at the University of Exeter. Appendix A provides a basic overview of the model, the physics it simulates and the numerical schemes.

Finally in chapter 7 we draw all these threads together to discuss the results in the context of astrophysical research, and how they could inform future studies as our observational and computational technologies improve.

2

_·

IDEALISED MODELS

The motions of the atmosphere are described by the Navier-Stokes fluid equations, and the laws of thermodynamics. These comprise a set of coupled, nonlinear partial differential equations that cannot be directly solved. There are two approaches to proceed:

1. Solve the equations numerically.

2. Reduce the equations to a simpler system.

To understand the dynamics of exoplanetary atmospheres, neither of these choices alone are entirely satisfactory. Using the biggest and most powerful supercomputers and discretising the domain, we can simulate the motion of the atmosphere to a reasonable level of fidelity. The MetOffice operational weather forecast, one of the most sophisticated models in terms of physical processes and interactions represented in an atmosphere, can, given sufficient processing power, compute the weather on surface of the Earth with a resolution on the order of 10 km2. The forecast may be accurate in telling us how the weather is; it is the result of hundreds of interactions all influencing the state and therein lies its limitation as a tool if we wish to gain an understanding of why the weather is. Once combined, the individual effects are inseparable from the net result. Furthermore, a complex model requires a complex initial state. We need to know the bulk conditions to get the climate correct – radiation balance, atmospheric composition, surface interactions; specific local observations across the surface and throughout the atmosphere are needed to get the weather right.

16 IDEALISED MODELS

Reducing the equations to a simpler set eliminates parameters; fewer interactions and fewer unknowns allows causal relationships to be drawn between physical process and climatology. However, in simplifying the equations we have to make assumptions about how to simplify. We must justify the removal or omission of physical processes in a consistent manner, otherwise the model is a model, but not necessarily a model of the atmosphere. And in simplifying we inevitably lose fidelity. A simple model may tell us why the jet stream exists, and has undulations over the United Kingdom, but it won’t tell us where they are at a given time.

All of this applies to exoplanetary atmospheres. For an exoplanet few, if any at all, of the input parameters will be known to a high degree of certainty, for every unknown we can make an educated guess, but we must also consider its impact on the larger climate if it varies. Or, we can apply a simpler model to eliminate it, and understand that this makes the conclusions drawn much more general.

In this thesis we apply a combination of both approaches, using models of various complexity to study the influence of diurnal timescales on the exoplanet dynamics. All of these models have their roots in the Navier–Stokes equations so we will begin there. By making assumptions on the scale and nature of the atmosphere, from this we derive a series of simpler systems. These will be used in the upcoming chapters to demonstrate the relationship between diurnal cycle and characteristic features of the exoplanet climatology.

2.1

Equations of motion

The dynamics of a dry, chemically well-mixed atmosphere will be described by a state vector comprising of velocity u = (u, v, w), temperature, T, pressure, p and fluid density,ρ. The domain considered will typically be a layer above the surface of a sphere of radius a, in spherical coordinates(z, λ, ϕ) shown in fig. 2.1, or a locally flat Cartesian domain(x, y, z).

2.1 EQUATIONS OF MOTION 17 φ r z x λ λ_{0 0} a r=a+z kˆ

Figure 2.1: The spherical coordinate systemused throughout this study. The radial compo-

nentrwill usually be expressed in terms ofz, the distance above the solid surface

of radiusa. Latitudeϕis zero at the equator and increases monotonically from the

South to the North Pole. Longitudeλis orientated relative to an origin angleλ0.

dimensions, Du Dt + f × u = 1 ρ∇p + ν∇ 2_{u+ g , (2.1)}

where f = 2Ω sin ϕˆk is the Coriolis vector, oriented along ˆk perpendicular to the surface, g is the gravitational vector (this can be considered effective gravity, the effects of centrifugal force absorbed into it) andν the kinematic viscosity. Momentum is subject to the constraints of conservation of mass, given by

Dρ

Dt + ρ∇ · u = 0, (2.2)

and the entropy of the system given by the thermodynamic equation

c_pDθ Dt =

θ

TQ˙, (2.3)

where c_pis the specific heat capacity of the atmosphere. External heating, the primary method of forcing that will be applied in the work presented here, is given by ˙Q, and it can be seen in this form that in the absence of any external heating or cooling,

18 IDEALISED MODELS

˙

Q= 0, the potential temperature,

θ = T p p0

‹R/cp

, (2.4)

for a reference pressure p0and gas constant R, is materially conserved. The system is closed by an equation of state, the ideal gas law,

p= ρRT. (2.5)

Standard values for the constants can be found in the List of Symbols and Constants.